Let $E$ be an elliptic curve which has CM by $ \Bbb{Q}( \sqrt{-5})$ with j invariant $63200+282880\sqrt{5}$

algebraic-number-theoryarithmetic-geometryelliptic-curves

Let $E$ be an elliptic curve which has CM by $ \Bbb{Z}[ \sqrt{-5}]$ with j invariant $632000+282880\sqrt{5}$.

I want to know whether $E$ has good reduction at $(2,1＋\sqrt{―5})$ or not, and to know what is $ \sharp E[ \Bbb{F_2}]$.
But I'm stuck with calculating this by hand.

I couldn't find this elliptic curve in LMGDB.

Thank you for your help.

Best Answer

Consider the following elements in the field $H=\Bbb Q(\sqrt{-1},\sqrt{-5})$: $$ \begin{aligned} k &= \sqrt{-1}\ ,\\ a &= \sqrt{-5}\ ,\\ ak &=\sqrt{5} \text{ (in a manner compatible with the choices for $a,k$)} \\[3mm] s &=\frac 12(\sqrt{-1}+\sqrt{-5})\ . \end{aligned} $$ Then $s$ is an algebraic integer, the minimal polynomial of $s$ is $F=X^4 + 3X^2 + 1$, so $s$ is of norm one, and $1,s,s^2,s^3$ turns out to be an integral basis of $H$. The reduction modulo two of $F$ is $f=x^4 + x^2 + 1=(x^2+x+1)^2$, so it is natural to consider $\Bbb F_4:=\Bbb F_2[u]$, where $u$ has minimal polynomial $(x^2+x+1)$. There is thus a map $\Phi$ from the ring $\mathcal O_H$ to $\Bbb F_4$ defined on the generator $s$ by $s\to u$.

In particular $\Phi(s^2)=u^2=u+1$.

The elements $a,k,ak$ have the following expressions in terms of $s$: $a=s^3+4s$, $k=s^3 +2s$, $ak=-2s^2-3$. We compute $\Phi(2)=0$, $\Phi(1+ak)=0$, so $\Phi$ induces a map from $\mathcal O_H$ modulo $(2,1\pm ak)$ (which corresponds to $(2,1+\sqrt 5)$ in either choice of the square root) to $\Bbb F_4$.

We will use $\Phi$ when passing from an equation defined over $\mathcal O_H$ to one over $\Bbb F_4$. (I.e. we apply the tensor functor $-\otimes_{\mathcal O_H}\Bbb F_4$ with $\Phi$ as structural map.)

Time to get the equation of the curve. Computers are in this century the weapon of choice, when a lookup in the database goes into the void. In the above theoretical setup, we use the code:

R.<x> = PolynomialRing(QQ)
K.<s> = NumberField( ( (sqrt(-5) + sqrt(-1))/2 ).minpoly() )
a = (x^2 + 5).roots(ring=K, multiplicities=False)[0]
k = (x^2 + 1).roots(ring=K, multiplicities=False)[0]
j = 320*(1975 + 884*a*k)
E = EllipticCurve_from_j(j)

And E is:

sage: E
Elliptic Curve defined by 
    y^2 = x^3 + (2142429020160*s^2+818331402240)*x 
              + (1803656541954375680*s^2+688935500534251520)
over Number Field in s with defining polynomial x^4 + 3*x^2 + 1

(Code was manually rearranged.)

Now the whole discussion about $\Phi$ is computationally useless, (but structurally needed to have a morphism to "something" in characteristic two, thus giving sense to the story,) since over $\Bbb F_2$ most involved coefficients vanish, giving rise to $y^2=x^3$.

$\square$

The answer is finished, but i tried to find other $j$-values (to search for them), those for which we may obtain after reduction regular elliptic curves over $\Bbb F_4$. The list of these possible CM-discriminants, that are not discarded from the start (having $a_4$, $a_6$ not zero modulo $(2)$) was obtained via:

sage: KK.<sqrt5> = QuadraticField(5)
sage: RZ.<Y> = PolynomialRing(ZZ)
sage: RF2.<X> = PolynomialRing(GF(2))

sage: for j in cm_j_invariants(K):
....:     F = RQ(j.minpoly())
....:     f = F(X)    # reduction modulo two
....:     if f.is_irreducible() and f.degree() == 2:
....:         print(f'j = {j} is among {F.roots(ring=KK, multiplicities=False)}')
....:         E = EllipticCurve_from_j(j)
....:         print(E, '\n')

We get:

j = 85995*s^2 + 33480 is among [85995/2*sqrt5 - 191025/2, -85995/2*sqrt5 - 191025/2]
Elliptic Curve defined by 
y^2 = x^3 + (49727382705*s^2+18996249195)*x
          + (-6410928347984160*s^2-2448758247517890)
over Number Field in s with defining polynomial x^4 + 3*x^2 + 1 

j = -85995*s^2 - 224505 is among [85995/2*sqrt5 - 191025/2, -85995/2*sqrt5 - 191025/2]
Elliptic Curve defined by 
y^2 = x^3 + (-49727382705*s^2-130185898920)*x 
          + (6410928347984160*s^2+16784026796434590)
over Number Field in s with defining polynomial x^4 + 3*x^2 + 1 

j = 16554983445*s^2 + 43341513480 is among [16554983445/2*sqrt5 + 37018076625/2, -16554983445/2*sqrt5 + 37018076625/2]
Elliptic Curve defined by 
y^2 = x^3 + (-1838500851256815240495*s^2-4813257716935987046805)*x 
          + (-45371836401785381434095138749760*s^2-118785009831873858847333508631390)
over Number Field in s with defining polynomial x^4 + 3*x^2 + 1 

j = -16554983445*s^2 - 6323436855 is among [16554983445/2*sqrt5 + 37018076625/2, -16554983445/2*sqrt5 + 37018076625/2]
Elliptic Curve defined by 
y^2 = x^3 + (1838500851256815240495*s^2+702244836834458674680)*x
          + (45371836401785381434095138749760*s^2+17330499373482285454951907617890)
over Number Field in s with defining polynomial x^4 + 3*x^2 + 1

And over $\Bbb F_4=\Bbb F_2[u]$ we get respectively the curves (recalling $\Phi(s^2)=u+1$) $E_1$, $E_2$, $E_3=E_1$, $E_4=E_2$, with: $$ \begin{aligned} E_1 &:\qquad y^2 = x^3 + ux\ ,\\ E_2 &:\qquad y^2 = x^3 + (u+1)x \ . \end{aligned} $$ They are singular because of $x^3 +ux=x(x+u+1)^2$, and $x^3+(u+1)x=x(x+u)^2$.

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