Finding an elliptic curve with CM by $\mathbb{Z}[\sqrt{-17}]$

Question

I have the imaginary quadratic field $K= \mathbb{Q}(\sqrt{-17})$ with $\mathcal{O}_K = \mathbb{Z}[\sqrt{-17}]$. Now I want to have the $j$-Invariant of an elliptc curve $E$ with complex multiplication by $\mathcal{O}_K$. For a given elliptic curve I know how to compute the $j$-invariant with PARI. Therefore just a hint how to get this curve would be perfect!

Answer 1

The class group $\operatorname{Cl}(\mathcal{O}_K)$ has order $4$ and consists of $ \langle 1 \rangle, \langle 2, 1+\sqrt{-17} \rangle, \langle 3, 1 + \sqrt{-17} \rangle$ and $\langle 3, 2 + \sqrt{-17} \rangle$.

One elliptic curve with CM by $\mathbb{Z}[\sqrt{-17}]$ is just $\mathbb{C}/\mathbb{Z}[\sqrt{-17}]$. From the action of $\operatorname{Cl}(\mathcal{O}_K)$ on elliptic curves with CM by $\mathcal{O}_K$ we get three more elliptic curves, given by lattices homothetic to $\mathbb{Z}\big[\frac{1+\sqrt{-17}}{2}\big]$, $\mathbb{Z}\big[\frac{1+\sqrt{-17}}{3}\big]$, and $\mathbb{Z}\big[\frac{2+\sqrt{-17}}{3}\big]$.

The $j$-invariant of $\mathbb{C}/\mathbb{Z}[\sqrt{-17}]$ is $j(\sqrt{-17})$ which is an algebraic integer of degree $\#\operatorname{Cl}(\mathcal{O}_K) = 4$ whose conjugates are exactly [notes, Theorem 11] the $j$-invariants of the elliptic curves above, i.e. $j\big(\frac{1+\sqrt{-17}}{2}\big), j\big(\frac{1+\sqrt{-17}}{3}\big)$ and $j\big(\frac{2+\sqrt{-17}}{3}\big)$.

The approximate values of these numbers can be found using PARI/GP's ellj:

? ellj(sqrt(-17))
%1 = 178211465582.57236317285989152242246715
? ellj((1+sqrt(-17))/2)
%2 = -421407.46393796828425027276471142244105
? ellj((1+sqrt(-17))/3)
%3 = -2087.5542126022878206248288778733953807 - 4843.8060029534518331212466414598790536*I
? ellj((2+sqrt(-17))/3)
%4 = -2087.5542126022878206248288778733953807 + 4843.8060029534518331212466414598790536*I

Now $j(\sqrt{-17})$ is a root of $f = \prod (x-j_k) \in \mathbb{Z}[x]$ where the $j_k$ are all the conjugates. We can approximate $f$ by replacing the $j_k$ with their numerical approximations and expanding the product. In this way we find that $j(\sqrt{-17})$ is very probably the unique positive real root of $$x^4 - 178211040000x^3 - 75843692160000000x^2 - 318507038720000000000x - 2089297506304000000000000,$$ namely $$8000 (5569095 + 1350704 \sqrt{17} + 4 \sqrt{3876889241278 + 940283755330 \sqrt{17}}),$$ which can be verified numerically to very high precision.

As mentioned in the comments we can also find an explicit Weierstrass form for an elliptic curve with given $j$-invariant.

Answer 2

To compute this sort of thing in Sage, you can use cm_j_invariants(). For example:

K.<a>=NumberField(x^2+17) H0.<b>=NumberField(K.hilbert_class_polynomial()) cm_j_invariants(H0)

The output is:

[-12288000,
 54000,
 0,
 287496,
 1728,
 16581375,
 -3375,
 8000,
 -32768,
 -884736,
 -884736000,
 -147197952000,
 -262537412640768000,
 2911923/16469087501000000000*b^3 - 1621681416684/51465898440625*b^2 - 6605556732936/50210632625*b - 850332053302272/968769853,
 -2911923/16469087501000000000*b^3 + 1621681416684/51465898440625*b^2 + 6605556732936/50210632625*b - 5367201819436127232/968769853,
 1124064022653/7750158824000000*b^3 - 626003413105165524/24219246325*b^2 - 2549884963677490296/23628533*b - 696625107656760443520000/968769853,
 -1124064022653/7750158824000000*b^3 + 626003413105165524/24219246325*b^2 + 2549884963677490296/23628533*b - 4402688233312810879896960000/968769853,
 b,
 -5041/886477376000000*b^3 + 701846857/692560450*b^2 + 58591882490/13851209*b + 22967913600000/814777]

The last number in the list is the right one:

j=-5041/886477376000000*b^3 + 701846857/692560450*b^2 + 58591882490/13851209*b + 22967913600000/814777

Now

j.minpoly()

identifies it as the root of:

x^4 - 178211040000*x^3 - 75843692160000000*x^2 - 318507038720000000000*x - 2089297506304000000000000

To test it, you can define a number field with that minimal polynomial, or alternatively

L.<b0>=H0.subfield(j)[0];L

which gives:

Number Field in b0 with defining polynomial x^4 - 178211040000*x^3 - 75843692160000000*x^2 - 318507038720000000000*x - 2089297506304000000000000

Then

E=EllipticCurve_from_j(b0);E

shows

Elliptic Curve defined by y^2 = x^3 + (-3*b0^2+5184*b0)*x + (-2*b0^3+6912*b0^2-5971968*b0) over Number Field in b0 with defining polynomial x^4 - 178211040000*x^3 - 75843692160000000*x^2 - 318507038720000000000*x - 2089297506304000000000000

Checking that it has the right CM order:

E.cm_discriminant()

outputs

-68