ideals in $\mathbb{Q}(\sqrt{-5})$ with norm less than $100$?

Question

Let $K=\mathbb{Z}(\sqrt{-5})$ be a field and $\mathcal{O}_K=\mathbb{Z}(\sqrt{-5})$ be the ring of integers. What are the ideals $\mathfrak{a} \subseteq \mathcal{O}_K=\mathbb{Z}(\sqrt{-5})$ with norm $N(\mathfrak{a})<100$? For a principal ideal $\mathfrak{a}=(a_1+a_2\sqrt{-5})$ the norm is $a_1^2+5a_2^2<100$ this region is bounded by an ellipse and so these ideals are straightforward to list. However, I know this field has class number two so this list must be incomplete.

Even a list of prime ideals might be more straightforward to write. We could list primes with $[p\mathbb{Z}(\sqrt{-5}):\mathbb{Z}(\sqrt{-5})]=p^2=100$ and some of these lattices will factor. However there are certainly other prime ideals $\mathfrak{p}\subseteq \mathbb{Z}(\sqrt{-5})$ that do not lie on the real number line $\mathbb{R}$.

On Wikipedia there's a fairly long list of Gaussian integers and their prime factorization in $\mathbb{Z}(i)$. However no such data set exists in this ring.

Answer 1

sage has an implementation for this. After defining K as below, one can ask details on the implementation via ?K.ideals_of_bdd_norm.

See also ideals_of_bdd_norm.

sage: K.<a> = QuadraticField(-5)
sage: K
Number Field in a with defining polynomial x^2 + 5
sage: for norm, ideals in K.ideals_of_bdd_norm(100).items():
....:     for J in ideals:
....:         print norm, J
....:         
1 Fractional ideal (1)
2 Fractional ideal (2, a + 1)
3 Fractional ideal (3, a + 2)
3 Fractional ideal (3, a + 1)
4 Fractional ideal (2)
5 Fractional ideal (-a)
6 Fractional ideal (-a + 1)
6 Fractional ideal (a + 1)
7 Fractional ideal (7, a + 4)
7 Fractional ideal (7, a + 3)
8 Fractional ideal (4, 2*a + 2)
9 Fractional ideal (-a - 2)
9 Fractional ideal (3)
9 Fractional ideal (a - 2)
10 Fractional ideal (10, a + 5)
12 Fractional ideal (6, 2*a + 4)
12 Fractional ideal (6, 2*a + 2)
14 Fractional ideal (a - 3)
14 Fractional ideal (a + 3)
15 Fractional ideal (15, a + 5)
15 Fractional ideal (15, a + 10)
16 Fractional ideal (4)
18 Fractional ideal (18, a + 11)
18 Fractional ideal (6, 3*a + 3)
18 Fractional ideal (18, a + 7)
20 Fractional ideal (-2*a)
21 Fractional ideal (2*a + 1)
21 Fractional ideal (a + 4)
21 Fractional ideal (a - 4)
21 Fractional ideal (-2*a + 1)
23 Fractional ideal (23, a + 15)
23 Fractional ideal (23, a + 8)
24 Fractional ideal (-2*a + 2)
24 Fractional ideal (2*a + 2)
25 Fractional ideal (5)
27 Fractional ideal (27, a + 20)
27 Fractional ideal (9, 3*a + 6)
27 Fractional ideal (9, 3*a + 3)
27 Fractional ideal (27, a + 7)
28 Fractional ideal (14, 2*a + 8)
28 Fractional ideal (14, 2*a + 6)
29 Fractional ideal (2*a + 3)
29 Fractional ideal (-2*a + 3)
30 Fractional ideal (-a - 5)
30 Fractional ideal (-a + 5)
32 Fractional ideal (8, 4*a + 4)
35 Fractional ideal (35, a + 25)
35 Fractional ideal (35, a + 10)
36 Fractional ideal (-2*a - 4)
36 Fractional ideal (6)
36 Fractional ideal (2*a - 4)
40 Fractional ideal (20, 2*a + 10)
41 Fractional ideal (a - 6)
41 Fractional ideal (a + 6)
42 Fractional ideal (42, a + 11)
42 Fractional ideal (42, a + 25)
42 Fractional ideal (42, a + 17)
42 Fractional ideal (42, a + 31)
43 Fractional ideal (43, a + 34)
43 Fractional ideal (43, a + 9)
45 Fractional ideal (2*a - 5)
45 Fractional ideal (-3*a)
45 Fractional ideal (2*a + 5)
46 Fractional ideal (-3*a + 1)
46 Fractional ideal (3*a + 1)
47 Fractional ideal (47, a + 29)
47 Fractional ideal (47, a + 18)
48 Fractional ideal (12, 4*a + 8)
48 Fractional ideal (12, 4*a + 4)
49 Fractional ideal (3*a - 2)
49 Fractional ideal (7)
49 Fractional ideal (-3*a - 2)
50 Fractional ideal (10, 5*a + 5)
54 Fractional ideal (a - 7)
54 Fractional ideal (-3*a + 3)
54 Fractional ideal (3*a + 3)
54 Fractional ideal (-a - 7)
56 Fractional ideal (2*a - 6)
56 Fractional ideal (2*a + 6)
58 Fractional ideal (58, a + 45)
58 Fractional ideal (58, a + 13)
60 Fractional ideal (30, 2*a + 10)
60 Fractional ideal (30, 2*a + 20)
61 Fractional ideal (3*a + 4)
61 Fractional ideal (-3*a + 4)
63 Fractional ideal (63, a + 11)
63 Fractional ideal (21, 3*a + 12)
63 Fractional ideal (63, a + 25)
63 Fractional ideal (63, a + 38)
63 Fractional ideal (21, 3*a + 9)
63 Fractional ideal (63, a + 52)
64 Fractional ideal (8)
67 Fractional ideal (67, a + 53)
67 Fractional ideal (67, a + 14)
69 Fractional ideal (2*a + 7)
69 Fractional ideal (a - 8)
69 Fractional ideal (a + 8)
69 Fractional ideal (-2*a + 7)
70 Fractional ideal (-3*a - 5)
70 Fractional ideal (3*a - 5)
72 Fractional ideal (36, 2*a + 22)
72 Fractional ideal (12, 6*a + 6)
72 Fractional ideal (36, 2*a + 14)
75 Fractional ideal (15, 5*a + 10)
75 Fractional ideal (15, 5*a + 5)
80 Fractional ideal (-4*a)
81 Fractional ideal (4*a - 1)
81 Fractional ideal (-3*a - 6)
81 Fractional ideal (9)
81 Fractional ideal (3*a - 6)
81 Fractional ideal (-4*a - 1)
82 Fractional ideal (82, a + 35)
82 Fractional ideal (82, a + 47)
83 Fractional ideal (83, a + 59)
83 Fractional ideal (83, a + 24)
84 Fractional ideal (4*a + 2)
84 Fractional ideal (2*a + 8)
84 Fractional ideal (2*a - 8)
84 Fractional ideal (-4*a + 2)
86 Fractional ideal (a - 9)
86 Fractional ideal (a + 9)
87 Fractional ideal (87, a + 74)
87 Fractional ideal (87, a + 16)
87 Fractional ideal (87, a + 71)
87 Fractional ideal (87, a + 13)
89 Fractional ideal (4*a - 3)
89 Fractional ideal (-4*a - 3)
90 Fractional ideal (90, a + 65)
90 Fractional ideal (30, 3*a + 15)
90 Fractional ideal (90, a + 25)
92 Fractional ideal (46, 2*a + 30)
92 Fractional ideal (46, 2*a + 16)
94 Fractional ideal (-3*a + 7)
94 Fractional ideal (3*a + 7)
96 Fractional ideal (-4*a + 4)
96 Fractional ideal (4*a + 4)
98 Fractional ideal (98, a + 81)
98 Fractional ideal (14, 7*a + 7)
98 Fractional ideal (98, a + 17)
100 Fractional ideal (10)
sage: 

Answer 2

You can consider how primes split in $\mathcal{O}_K=\mathbb{Z}(\sqrt{-5})$. If we try to find the discriminant since $-5 \equiv 3\pmod 4$, $D= 4 \cdot -5=-20$, and primes not dividing $-20$ split if $\left(\frac{-20}{p}\right)=\left(\frac{-4}{p}\right)\left(\frac{p}{5}\right)=1$. From there you should be able to get a set of congruence conditions for primes that split. As far as primes that don't split, you only need consider those primes $<10$, since otherwise $N(p)=p^2>100$. Finally $2$ and $5$ both ramify so primes of $\mathcal{O}_K$ lying above them also need to be considered but should be a short, finite check from there.

I get $\{2, 3, 5, 7, 23, 29, 37, 41, 43, 47, 61, 67, 83, 89\}$ as the list of primes with norm less than $100$ that ramify or split using the congruences $p \equiv 1, 3, 7, 9 \pmod {20}$ for primes that split in $\mathcal{O}_K$. There are no inert primes with norm $<100$.