Finding a point on a Fermat related curve

Question

I am trying to find solutions to the equation $ w^5+x^5=y^5+z^5 $. To do this I am performing a substitution $w=r+\sqrt{s}, x=r-\sqrt{s}, y=t+\sqrt{u}, z=t-\sqrt{u}$. This results in the equation: $r^5+10r^3s+5rs^2=t^5+10t^3u+5tu^2$.

After trying various values of r and t I stumbled upon a promising candidate: $r=1, t=-4$. This results in the equation $(s+1)^2+(2u+32)^2=820$, which has solutions in s and u.

I want s to be a square, which will yield a near solution to the original equation (albeit still involving $\sqrt{u}$). Setting $x=s^2+1, y=2u+32$ results in the curve $y^2 +(x^2+1)^2=820$. This would be an elliptic curve, if there could be found a rational point on it.

So my question is: Does the curve $y^2 +(x^2+1)^2=820$ have a rational point?

Edit: The first test case had an obvious solution that I missed. This resulted in the nice equation: $6^5-4^5=(-4-\sqrt{-10})^5+(-4+\sqrt{-10})^5$. As player3236 pointed out, no value of u in this equation will be positive, so the solutions will always involve an imaginary component. This will happen any time t is chosen to be negative. However, if t is positive there is potential for u to also be positive.

If r=1, t=16, the new equation is $(x^2+1)^2+838860=y^2$. Does this new equation have rational solutions? Wolfram Alpha couldn't find any this time.

Edit 2: I did find a point on the curve with $x=\frac{931}{34}$

Answer 1

Let us consider the first curve, $y^2+(x^2+1)^2= 820$. Finding points on this curve is equivalent to finding points on the curve $$ \begin{aligned} E'\ &:\ &V^2 &= -((U+5)^2+1)^2+820\ ,\text{ i.e.}\\ E'\ &:\ &V^2 &= -U^4 - 20U^3 - 152U^2 - 520U + 144\ . \end{aligned} $$ Let us write the above in the form $V^2 =aU^4+bU^3+cU^2+dU+q^2$, with obvious coefficients $a,b,c,d; q$. Let $Q$ be $Q=2q$. Now Ian Connell, Elliptic Curve Handbook, Proposition 1.2.1, page 105 gives a birational passage to the elliptic curve with equation $$ E''\ :\ Y^2+a_1XY+a_3Y=X^3+a_2X^2+a_4X+a_6\ ,$$ where: $$ \begin{aligned} a_1 &= d/q\ ,\\ a_2 &= c-d^2/Q^2\ ,\\ a_3 &= Qb\ ,\\ a_4 &= -Q^2a\ ,\\ a_6 &= a_2a_4 = a(d^2-Q^2c)\ . \end{aligned} $$ A point $(X,Y)$ on the curve $E''$ induces a point $(U,V)$ on $E'$ by the formulas: $$ U = \frac 1Y\left(Q(X+c)-\frac 1Q\cdot d^2\right)\ ,\qquad V = -q+\frac 1Q\cdot U(UX-d)\ . $$ In our case, the curve $E''$ is the following curve, sage code:

sage: var('x,y');
sage: def f(x,y):    return -y^2 - (x^2+1)^2 + 820
sage: f(x+5, y).expand()
-x^4 - 20*x^3 - 152*x^2 - y^2 - 520*x + 144
sage: a, b, c, d, q = -1, -20, -152, -520, 12
sage: Q = 2*q
sage: a1, a2, a3, a4 = d/q, c-d^2/Q^2, Q*b, -Q^2*a
sage: a6 = a2*a4
sage: E = EllipticCurve(QQ, [a1, a2, a3, a4, a6])
sage: E.rank()
2
sage: E.torsion_points()
[(0 : 1 : 0), (52 : 4100/3 : 1)]

This gives a lot of points on the curve. The following code constructs some of them:

solutions = []
P1, P2 = E.gens()
T = E.torsion_points()[1]
for j1 in [-3..3]:
    for j2 in [-3..3]:
        for n in [0..1]:
            R = j1*P1 + j2*P2 + n*T
            if R == E(0):    continue
            X, Y = R.xy()
            if Y == 0:    continue
            U = (Q*(X+c) - d^2/Q) / Y
            V = -q + U*(U*X - d)/Q
            U5, V = abs(U+5), abs(V)
            if (U5, V) in solutions:    continue
            solutions.append( (U5, V) )
            u = V/2 - 16
            # if u < 0:    continue
            print(r"x &= \frac{%s}{%s}\\"
                  % (U5.numer(), latex(factor(U5.denom()))))
            print(r"y &= \frac{%s}{%s}\\"
                  % (V.numer(), latex(factor(V.denom()))))
            print(r"u &= \frac{%s}{%s}\\"
                  % (latex(factor(u.numer())), latex(factor(u.denom()))))
            if u.is_square():    print(r"&SQUARE\\")
            print("  &\\qquad y^2 + (x^2+1)^2 - 820 = %s\n\\\\[4mm]\n%%"
                  % f(U5, V))

We have then the following solutions: $$ \begin{aligned} x &= \frac{13036658315510032988204253801235}{13 \cdot 29 \cdot 37 \cdot 21701 \cdot 62233 \cdot 87697 \cdot 5563471867729}\\ y &= \frac{2405690869878559044594514807781718914294941723532442030205314708}{13^{2} \cdot 29^{2} \cdot 37^{2} \cdot 21701^{2} \cdot 62233^{2} \cdot 87697^{2} \cdot 5563471867729^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 11 \cdot 23 \cdot 39733 \cdot 1952241098341 \cdot 3791563425833816372596018279215226374188039}{13^{2} \cdot 29^{2} \cdot 37^{2} \cdot 21701^{2} \cdot 62233^{2} \cdot 87697^{2} \cdot 5563471867729^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{162458182555112323161}{5 \cdot 97 \cdot 73623006391638001}\\ y &= \frac{23822580678926521285740950989704151999428}{5^{2} \cdot 97^{2} \cdot 73623006391638001^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 13 \cdot 71 \cdot 333589 \cdot 2236160247547007 \cdot 6164477497937567}{5^{2} \cdot 97^{2} \cdot 73623006391638001^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{9520442060365}{3642799992361}\\ y &= \frac{365511913152330277363122252}{3642799992361^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 5 \cdot 103 \cdot 28702827158601638691367}{3642799992361^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{230878603791}{5 \cdot 6829 \cdot 1874317}\\ y &= \frac{102280146333948447788892}{5^{2} \cdot 6829^{2} \cdot 1874317^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 13 \cdot 29 \cdot 2998141 \cdot 6366883182061}{5^{2} \cdot 6829^{2} \cdot 1874317^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{93542279748835}{27917 \cdot 898599293}\\ y &= \frac{15387605729678347093142160468}{27917^{2} \cdot 898599293^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 13 \cdot 23 \cdot 683 \cdot 1559 \cdot 553099907 \cdot 6744379151}{27917^{2} \cdot 898599293^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{4992964600942606254201}{5 \cdot 8233 \cdot 186581 \cdot 262351875749}\\ y &= \frac{112597556643703829003523268375206881581624068}{5^{2} \cdot 8233^{2} \cdot 186581^{2} \cdot 262351875749^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 71 \cdot 687289 \cdot 35949192861661 \cdot 2470089037887599849437}{5^{2} \cdot 8233^{2} \cdot 186581^{2} \cdot 262351875749^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{14100446200928414271395695884163165}{10133 \cdot 9131338913 \cdot 32850086210006457589}\\ y &= \frac{163409560140746617280818877377336839053725151518595253277111416731532}{10133^{2} \cdot 9131338913^{2} \cdot 32850086210006457589^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 5 \cdot 13^{2} \cdot 131 \cdot 40253120493723100804553179 \cdot 7419158906364631246083380484383646761}{10133^{2} \cdot 9131338913^{2} \cdot 32850086210006457589^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{61101860425670949880707892899}{5^{2} \cdot 265205909 \cdot 1913532975314529661}\\ y &= \frac{2465433439082350072102436786861216960772068168407137748332}{5^{4} \cdot 265205909^{2} \cdot 1913532975314529661^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 19 \cdot 9348263 \cdot 115636877949077 \cdot 18964570146285293 \cdot 1723484690850703801}{5^{4} \cdot 265205909^{2} \cdot 1913532975314529661^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{3182700450859025}{22129 \cdot 147875300509}\\ y &= \frac{305926196130319251970804940340252}{22129^{2} \cdot 147875300509^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 5 \cdot 43 \cdot 131 \cdot 5527 \cdot 58995650061058355531771}{22129^{2} \cdot 147875300509^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{640372149}{5^{2} \cdot 409 \cdot 11969}\\ y &= \frac{57258245706223668}{5^{4} \cdot 409^{2} \cdot 11969^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 105506277038649083}{5^{4} \cdot 409^{2} \cdot 11969^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{33025}{37 \cdot 3673}\\ y &= \frac{528512197188}{37^{2} \cdot 3673^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 23 \cdot 983 \cdot 691079}{37^{2} \cdot 3673^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{108983619}{5^{2} \cdot 850529}\\ y &= \frac{3950171837892948}{5^{4} \cdot 850529^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 11 \cdot 13^{2} \cdot 134581 \cdot 10509997}{5^{4} \cdot 850529^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{141166512970975}{13 \cdot 29 \cdot 90641 \cdot 2835253}\\ y &= \frac{267192592960622599823692451292}{13^{2} \cdot 29^{2} \cdot 90641^{2} \cdot 2835253^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 5 \cdot 103 \cdot 871062163457 \cdot 18493060659611}{13^{2} \cdot 29^{2} \cdot 90641^{2} \cdot 2835253^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{300465457943068995736805109}{5^{2} \cdot 548139169 \cdot 4846452536454329}\\ y &= \frac{83584824775100492091781590892844195239295177451886092}{5^{4} \cdot 548139169^{2} \cdot 4846452536454329^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 349 \cdot 214147 \cdot 5094064139 \cdot 20327786573 \cdot 1859326778033444504699597}{5^{4} \cdot 548139169^{2} \cdot 4846452536454329^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{2665287830263248801195859415}{113 \cdot 5472031792894910653554197}\\ y &= \frac{7989458932698213513688381614156914391236133060029901852}{113^{2} \cdot 5472031792894910653554197^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 5 \cdot 127 \cdot 1023257 \cdot 3429708645773522945507 \cdot 476274917191165521198937}{113^{2} \cdot 5472031792894910653554197^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{201902890715811}{5 \cdot 4729 \cdot 2901654253}\\ y &= \frac{126894617470983770240467695828}{5^{2} \cdot 4729^{2} \cdot 2901654253^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 93166237 \cdot 4886427649 \cdot 13035867311}{5^{2} \cdot 4729^{2} \cdot 2901654253^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{504185}{152821}\\ y &= \frac{608447007108}{152821^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 131 \cdot 229 \cdot 1157449}{152821^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{261}{5 \cdot 13}\\ y &= \frac{96972}{5^{2} \cdot 13^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 19 \cdot 503}{5^{2} \cdot 13^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{1415}{661}\\ y &= \frac{12271452}{661^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 5 \cdot 13 \cdot 6577}{661^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{5260848291}{5 \cdot 217539397}\\ y &= \frac{17744529175451385108}{5^{2} \cdot 217539397^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 11 \cdot 20939 \cdot 2804953 \cdot 7783379}{5^{2} \cdot 217539397^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{46203128231239359785}{37 \cdot 1337256503359283713}\\ y &= \frac{69953560999420955768703715813623276985668}{37^{2} \cdot 1337256503359283713^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 13 \cdot 1187 \cdot 135867929869169674775218266100668241}{37^{2} \cdot 1337256503359283713^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{5659195370905602649178527671}{5 \cdot 249389877977 \cdot 7234008048184553}\\ y &= \frac{2327274688594258349538778705315778236547654610669652515588}{5^{2} \cdot 249389877977^{2} \cdot 7234008048184553^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 11 \cdot 13^{2} \cdot 23 \cdot 4446191 \cdot 4885823 \cdot 11581429 \cdot 300089788549 \cdot 21414576369482434643}{5^{2} \cdot 249389877977^{2} \cdot 7234008048184553^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{218157575240795}{13 \cdot 29 \cdot 110077841033}\\ y &= \frac{352116840259517020019969772}{13^{2} \cdot 29^{2} \cdot 110077841033^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 5 \cdot 1559 \cdot 58909 \cdot 29811990671960835851}{13^{2} \cdot 29^{2} \cdot 110077841033^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{30159}{5 \cdot 37 \cdot 277}\\ y &= \frac{75115465692}{5^{2} \cdot 37^{2} \cdot 277^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 19 \cdot 103 \cdot 419 \cdot 2719}{5^{2} \cdot 37^{2} \cdot 277^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{5}{1}\\ y &= \frac{12}{1}\\ u &= \frac{-1 \cdot 2 \cdot 5}{1}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{9}{5}\\ y &= \frac{708}{5^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 23}{5^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{1850795}{432241}\\ y &= \frac{3946471955028}{432241^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 13 \cdot 911 \cdot 42897937}{432241^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{25093501670280879}{5 \cdot 97 \cdot 857 \cdot 15377 \cdot 1317793}\\ y &= \frac{1906726072369386038223938515453212}{5^{2} \cdot 97^{2} \cdot 857^{2} \cdot 15377^{2} \cdot 1317793^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 11446381 \cdot 7935036462375784894868837}{5^{2} \cdot 97^{2} \cdot 857^{2} \cdot 15377^{2} \cdot 1317793^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{10812125034210473313993622054805}{23072673329 \cdot 143581662104612874209}\\ y &= \frac{287075176351657878630882446552860447116555520044022707438613452}{23072673329^{2} \cdot 143581662104612874209^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 5 \cdot 13 \cdot 503 \cdot 10753 \cdot 559223123 \cdot 13447826686304359 \cdot 6062603250107171605870871959}{23072673329^{2} \cdot 143581662104612874209^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{1705233582536073760995190741133872101}{5 \cdot 244327342969 \cdot 266515540930558827574357}\\ y &= \frac{362431472808647895360320065190973444447130394211867699577642447822619532}{5^{2} \cdot 244327342969^{2} \cdot 266515540930558827574357^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 13 \cdot 29 \cdot 45932966323 \cdot 794963514281 \cdot 91794463398902240064343 \cdot 599400942698438604785969}{5^{2} \cdot 244327342969^{2} \cdot 266515540930558827574357^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % x &= \frac{3154741859903558524046951461789436053287025}{613 \cdot 359837 \cdot 709913 \cdot 7599836289694012095782190757}\\ y &= \frac{38930269299271168229289261505326698150375319537776856686256422554566753985971200683908}{613^{2} \cdot 359837^{2} \cdot 709913^{2} \cdot 7599836289694012095782190757^{2}}\\ u &= \frac{-1 \cdot 2 \cdot 23 \cdot 71 \cdot 33073 \cdot 77591 \cdot 159469 \cdot 96643537 \cdot 37692226417 \cdot 175669559711 \cdot 3736240723150976619930114531647866139}{613^{2} \cdot 359837^{2} \cdot 709913^{2} \cdot 7599836289694012095782190757^{2}}\\ &\qquad y^2 + (x^2+1)^2 - 820 = 0 \\[4mm] % \end{aligned} $$

---

On the second curve, $$ y^2+(x^2+1) = 838860\ , $$ we can more or less do the same. First, the curve is birational equivalent to...

var('x,y');
def f(x,y):    return -y^2 + (x^2+1)^2 + 838860
f(x + 931/34, y).expand()
a, b, c, d, q = 1, 1862/17, 2601439/578, 808030727/9826, 1369043/1156
Q = 2*q
a1, a2, a3, a4 = d/q, c-d^2/Q^2, Q*b, -Q^2*a
a6 = a2*a4
E = EllipticCurve(QQ, [a1, a2, a3, a4, a6])
rk = E.rank()
print("E is an elliptic curve of rank %s" % rk)
print("Generators:")
for P in E.gens():
    print("\t", P.xy())
for T in E.torsion_points():
    if T == E(0):    continue
    print("\t", T.xy())

And we get again a curve of rank two!

E is an elliptic curve of rank 2
Generators:
     (-1369043/578, 0)
     (-43316325/28322, -41113249906388/614068441)
     (-867917/578, -106213097760/1369043)

A better form is...

sage: E.minimal_model()
Elliptic Curve defined by y^2 = x^3 - x^2 - 209715*x over Rational Field

Then some points on $y^2 = (x^2+1)^2 + 838860 are:

 solutions = []
 P1, P2 = E.gens()
 T = E.torsion_points()[1]
 for j1 in [-2..2]:
     for j2 in [-2..2]:
         for n in [0..1]:
             R = j1*P1 + j2*P2 + n*T
             if R == E(0):    continue
             X, Y = R.xy()
             if Y == 0:    continue
             U = (Q*(X+c) - d^2/Q) / Y
             V = -q + U*(U*X - d)/Q
             UU, V = abs(U + 931/34), abs(V)
             if (UU, V) in solutions:    continue
             solutions.append( (UU, V) )
             print(r"x &= \frac{%s}{%s}\\"
                   % (UU.numer(), UU.denom()))
             print(r"y &= \frac{%s}{%s}\\"
                   % (V.numer(), V.denom()))
             print("  &\\qquad (x^2+1)^2 + 838860 - y^2 = %s\n\\\\[4mm]\n%%"
                   % ((UU^2+1)^2 + 838860 - V^2))

$$ \begin{aligned} x &= \frac{4640129248052012171295041}{56328368752930789692104}\\ y &= \frac{21729172452104012184427503136290818755933810040513}{3172885126366149767519655678919343059119946816}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{312505635338264918831323}{5635665178453622365027}\\ y &= \frac{101930516183122284005580603951145069243469807068}{31760722003634699217295111109664688832710729}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{7697107706380753981821512221}{186853640262015669462073894}\\ y &= \frac{67355326049228509693898695429792231527200858728844325683}{34914282879166763220702909129407082141202455516323236}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{275273488143971163955319670170828849327}{8630446951340393789955958138130027033}\\ y &= \frac{102015582307789500427230911928915609414149466194469828884544309431564059167628}{74484614579900697494445334607256438244445639280727623918729596137310783089}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{3533796460790528465936090156525696152253306912169010481}{142147419283298414256273589168805617982980986867368204}\\ y &= \frac{22336879382547877312447713567495156125129879126864659964611288771826011654660047981168198612554244346982363633}{20205888808901837866902182036543346606858038621737120342109726603597659188209906227548837538265889310185616}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{66981948434235}{4493946651194}\\ y &= \frac{19038090760503750938241601891}{20195556503777767101625636}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{9827459237}{1028878515}\\ y &= \frac{974459905490087541356}{1058590998628605225}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{369725522175}{86671374244}\\ y &= \frac{6881632670450531112199441}{7511927113343506571536}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{10885438126521930467}{10771530642536438565}\\ y &= \frac{106267543905924773766136574076877880671436}{116025872383101461045183783794019259225}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{3106512218127349796388060274811685}{494034186048468200762637842991614}\\ y &= \frac{223760671475152068058777277411257298563715773166193274657640742778051}{244069776984572492223657898435483022217396770912501687097274324996}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{3382515311}{218686124}\\ y &= \frac{45283088549214408113}{47823620830143376}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{3017}{143}\\ y &= \frac{20832748}{20449}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{408358573}{11636677}\\ y &= \frac{207929610753260668}{135412251602329}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{64609139182230553631}{1409763762048977264}\\ y &= \frac{4555779364380348768652797926075775093953}{1987433864786485387900613108388925696}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{544306536341}{6280755390}\\ y &= \frac{298503660402510832364131}{39447888269014052100}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{1215}{7}\\ y &= \frac{1476956}{49}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % x &= \frac{931}{34}\\ y &= \frac{1369043}{1156}\\ &\qquad (x^2+1)^2 + 838860 - y^2 = 0 \\[4mm] % \end{aligned} $$

Note: I'll try to look closer, and see why we get gurves of rank two...