An elementary way to categorize these five particular types of elliptic curves

Question

This question has to do with five specific types of elliptic curves

1. Let $A$ be the set of all elliptic curves which can be affinely transformed through a composition of some rotation and some translation into the zero set of $P_1(x)-y^2$ where $P_1(x)$ is a third-degree polynomial with real coefficients and two non-real roots such that the zero set of $P_1(x)-y$ has an inflection point in the upper half-plane and the tangent line at this inflection point has the same end behavior as the zero set of $P_1(x)+y$

2. Let $B$ be the set of all elliptic curves which can be affinely transformed through a composition of some rotation and some translation into the zero set of $P_2(x)-y^2$ where $P_2(x)$ is a third-degree polynomial with real coefficients and two non-real roots such that the zero set of $P_2(x)-y$ has an inflection point in the upper half-plane and the tangent line at this inflection point has slope zero

3. Let $C$ be the set of all elliptic curves which can be affinely transformed through a composition of some rotation and some translation into the zero set of $P_3(x)-y^2$ where $P_3(x)$ is a third-degree polynomial with real coefficients and two non-real roots such that the zero set of $P_3(x)-y$ has an inflection point in the upper half-plane and the tangent line at this inflection point has the same end behavior as the zero set of $P_3(x)-y$

4. Let $D$ be the set of all elliptic curves which can be affinely transformed through a composition of some rotation and some translation into the zero set of $P_4(x)-y^2$ where $P_4(x)$ is a third-degree polynomial with real coefficients and two non-real roots such that the zero set of $P_4(x)-y$ has an inflection point on the $x$-axis

5. Let $E$ be the set of all elliptic curves which can be affinely transformed through a composition of some rotation and some translation into the zero set of $P_5(x)-y^2$ where $P_5(x)$ is a third-degree polynomial with real coefficients and two non-real roots such that the zero set of $P_5(x)-y$ has an inflection point in the lower half-plane

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Let $a,c$ $\in$ $(-\infty,0)$ $\cup$ $(0,\infty)$

Let $b$ $\in$ $ℝ$

Let $\alpha$ $\in$ $(0,\pi)$

Let $f(x)=a\left(x-b\right)\left(x^{2}-2c\cos\left(\alpha\right)x+c^{2}\right)$

Since any elliptic curve in $A\cup B\cup C\cup D\cup E$ can be affinely transformed through a composition of some rotation and some translation into the zero set of $g(x)-y^2$ where $g(x)$ is a third-degree polynomial with real coefficients and two non-real roots, any elliptic curve in $A\cup B\cup C\cup D\cup E$ can be affinely transformed through a composition of some rotation and some translation into the zero set of

$$a(x-b)(x-ce^{i\alpha})(x-ce^{-i\alpha})-y^2$$

or, equivalently, the zero set of

$$a\left(x-b\right)\left(x^{2}-2c\cos\left(\alpha\right)x+c^{2}\right)-y^2$$

Note that $\alpha$ is the argument of one of the non-real roots of $f(x)$ while $-\alpha$ is the argument of the other non-real root of $f(x)$

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Problem:

Determine what the conditions on $a,b,c$ and $α$ must be in order for the zero set of $f(x)−y^2$ to be an affine transformation - via translation and rotation only - of a curve belonging to $A$. Then do the same for $B,C,D$ and $E$.

I seek verification that statements 1,2,3,4 and 5 below are true and that my six paragraphs of justification for those statements comprise a valid justification for the five statements.

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My solution:

One can sort any elliptic curve in $A\cup B\cup C\cup D\cup E$ into exactly one of the disjoint sets $A,B,C,D$ and $E$ by using the following statements:

1. Any elliptic curve which can be affinely transformed through a composition of some rotation and some translation into the zero set of $a\left(x-b\right)\left(x^{2}-2c\cos\left(\alpha\right)x+c^{2}\right)-y^2$ is a member of $A$ if and only if $a(c\cos(\alpha)-b)>0$ and $af'(\frac{b+2c\cos(\alpha)}{3})<0$
2. Any elliptic curve which can be affinely transformed through a composition of some rotation and some translation into the zero set of $a\left(x-b\right)\left(x^{2}-2c\cos\left(\alpha\right)x+c^{2}\right)-y^2$ is a member of $B$ if and only if $a(c\cos(\alpha)-b)>0$ and $f'(\frac{b+2c\cos(\alpha)}{3})=0$
3. Any elliptic curve which can be affinely transformed through a composition of some rotation and some translation into the zero set of $a\left(x-b\right)\left(x^{2}-2c\cos\left(\alpha\right)x+c^{2}\right)-y^2$ is a member of $C$ if and only if $a(c\cos(\alpha)-b)>0$ and $af'(\frac{b+2c\cos(\alpha)}{3})>0$
4. Any elliptic curve which can be affinely transformed through a composition of some rotation and some translation into the zero set of $a\left(x-b\right)\left(x^{2}-2c\cos\left(\alpha\right)x+c^{2}\right)-y^2$ is a member of $D$ if and only if $b=c\cos(\alpha)$
5. Any elliptic curve which can be affinely transformed through a composition of some rotation and some translation into the zero set of $a\left(x-b\right)\left(x^{2}-2c\cos\left(\alpha\right)x+c^{2}\right)-y^2$ is a member of $E$ if and only if $a(c\cos(\alpha)-b)<0$

To prove that the above five statements are true:

• By definition, any curve in $D$ can be translated and rotated to become a curve which is the zero set of $f(x)-y^2$ such that the zero set of $f(x)-y$ has an inflection point on the $x$-axis. It can be easily checked that the inflection point of the zero set of $f(x)-y$ is at $(\frac{b+2c\cos(\alpha)}{3},f(\frac{b+2c\cos(\alpha)}{3}))$. It can easily be checked that $f(\frac{b+2c\cos(\alpha)}{3})=0$ if and only if $c\cos(\alpha)-b=0$
• By definition, any curve in $E$ can be translated and rotated to become a curve which is the zero set of $f(x)-y^2$ such that the zero set of $f(x)-y$ has an inflection point in the lower half-plane. It can be easily checked that the inflection point of the zero set of $f(x)-y$ is at $(\frac{b+2c\cos(\alpha)}{3},f(\frac{b+2c\cos(\alpha)}{3}))$. It can easily be checked that $f(\frac{b+2c\cos(\alpha)}{3})<0$ if and only if $a(c\cos(\alpha)-b)<0$
• By definition, any curve in $A\cup B\cup C$ can be translated and rotated to become a curve which is the zero set of $f(x)-y^2$ such that the zero set of $f(x)-y$ has an inflection point in the upper half-plane. It can be easily checked that the inflection point of the zero set of $f(x)-y$ is at $(\frac{b+2c\cos(\alpha)}{3},f(\frac{b+2c\cos(\alpha)}{3}))$. It can easily be checked that $f(\frac{b+2c\cos(\alpha)}{3})>0$ if and only if $a(c\cos(\alpha)-b)>0$
• By definition, any curve in $A$ can be translated and rotated to become a curve which is the zero set of $f(x)-y^2$ such that the tangent line at the inflection point of the zero set of $f(x)-y$ has the same end behavior as the zero set of $f(x)+y$. Since the zero set of $f(x)+y$ has the same end behavior as $ax^3+y$, it also has the same end behavior as $ax+y$. Thus, the slope of the tangent line at the inflection point must be of the same sign as $-a$. It can be easily checked that the inflection point of the zero set of $f(x)-y$ is at $(\frac{b+2c\cos(\alpha)}{3},f(\frac{b+2c\cos(\alpha)}{3}))$, which means $-af'(\frac{b+2c\cos(\alpha)}{3})>0$, which means $af'(\frac{b+2c\cos(\alpha)}{3})<0$
• By definition, any curve in $B$ can be translated and rotated to become a curve which is the zero set of $f(x)-y^2$ such that the tangent line at the inflection point of the zero set of $f(x)-y$ has slope zero. It can be easily checked that the inflection point of the zero set of $f(x)-y$ is at $(\frac{b+2c\cos(\alpha)}{3},f(\frac{b+2c\cos(\alpha)}{3}))$, which means $f'(\frac{b+2c\cos(\alpha)}{3})=0$
• By definition, any curve in $C$ can be translated and rotated to become a curve which is the zero set of $f(x)-y^2$ such that the tangent line at the inflection point of the zero set of $f(x)-y$ has the same end behavior as the zero set of $f(x)-y$. Since the zero set of $f(x)-y$ has the same end behavior as $ax^3-y$, it also has the same end behavior as $ax-y$. Thus, the slope of the tangent line at the inflection point must be of the same sign as $a$. It can be easily checked that the inflection point of the zero set of $f(x)-y$ is at $(\frac{b+2c\cos(\alpha)}{3},f(\frac{b+2c\cos(\alpha)}{3}))$, which means $af'(\frac{b+2c\cos(\alpha)}{3})>0$