Computing elliptic curves over $\mathbb{Q}(\sqrt{-11})$ with good ordinary reduction at $\mathfrak{p}\mid 3$ in Sage

algebraic-number-theorycomputational mathematicselliptic-curvesnumber theorysagemath

Let $K=\mathbb{Q}(\alpha)$ for $\alpha=\sqrt{-11}$. Since $-11 \equiv 1\pmod 3$ is a square, there is a factorization:
$$ 3\mathcal{O}_K=((1/2)\alpha-(1/2))((1/2)\alpha+(1/2)) = \mathfrak{p}_1\mathfrak{p}_2 $$

By Theorem 4.1 of Silverman's AEC:

Theorem 4.1: Let $E/\mathbb{F}_3$ be an elliptic curve given by a Weierstrass equation $$ E:y^2 = f(x) $$ where $f(x)\in \mathbb{F}_3[x]$ is a cubic polynomial with distinct roots in an algebraic closure. Then $E$ is supersingular if and only if the coefficient of $x^2$ is zero in $\mathbb{F}_3$.

Thus, I expect most elliptic curves $E/K$ with good reduction to be ordinary when reduced at either $\mathfrak{p}_1$ or $\mathfrak{p}_2$. Therefore, to verify this, I ran the following snippet over a large set of elliptic curves $E/K$ defined by $y^2 = x^3 + Ax+B$ with bounded coefficients $A,B\in\mathcal{O}_K$ such that $\text{max}(|A|^4,|B|^6)\leq N$ for some fixed $N$.

K.<a> = NumberField(x^2+11)
v1 = K.ideal(3).factor()[0][0]
v2 = K.ideal(3).factor()[1][0]

...

total_count = 0

good_at_v1 = 0
good_at_v2 = 0
good_at_v1_and_v2 = 0

good_ordinary_at_v1 = 0
good_ordinary_at_v2 = 0
good_ordinary_at_v1_and_v2 = 0

for (A,B) in some list:

    ...

    E = EllipticCurve([A,B])
    da1 = E.local_data(v1)
    da2 = E.local_data(v2)

    gv1 = da1.has_good_reduction()
    gv2 = da2.has_good_reduction()
        
    ov1 = False
    ov2 = False
        
    if gv1:
        good_at_v1 = good_at_v1 + 1
        ov1 = True if da1.minimal_model().reduction(v1).is_ordinary() else False
    if gv2:
        good_at_v2 = good_at_v2 + 1
        ov2 = True if da2.minimal_model().reduction(v2).is_ordinary() else False
    if gv1 and gv2:
        good_at_v1_and_v2 = good_at_v1_and_v2 + 1

    if ov1 and ov2:
        good_ordinary_at_v1_and_v2 = good_ordinary_at_v1_and_v2 + 1
        
    total_count = total_count + 1

At the end of this code sequence, I get the following statistic:

  • $~65.8\%$ curves have good reduction at v1
  • $~65.8\%$ curves have good reduction at v2
  • $~44.7\%$ curves have good reduction at both v1 and v2
  • $0$ curves have good and ordinary reduction at v1
  • $0$ curves have good and ordinary reduction at v2

There are at least $50,000$ curves in my list, everything up to $N=200$. I'm baffled because it does not seem possible to get literally zero curves with good and ordinary reduction at either prime dividing $3$. Does anyone spot any obvious mistakes in my understanding, or my code? Something that I noticed is that for all of the Weierstrass equations in my list, .minimal_model() returns the exact same equation, and hence (by Theorem 4.1) every equation of the form $y^2 = x^3 + Ax+ B$ reduces to a supersingular curve (if it has good reduction). I have tried to use .minimal_model(reduce=False) but it returns the same results.

I hope this question is sufficiently mathematical to belong on math.SE. If not, please direct me to another active community where I should post this question instead. Thanks!

Edit 1. As @Mindlack points out in the comments, $E = X_0(11)$ has good ordinary reduction at $3$.
$$ E: y^2 + y = x^3 – x^2 – 10x – 20 $$
It has short Weierstrass form $E': y^2 = x^3 -13392x-1080432$. Using Sage, I found the minimal models at v1 of both equations.
$$ \begin{align*}
E &: y^2 + y = x^3 – x^2 – 10x – 20 \tag{did not change} \\
E' &: y^2 = x^3 + (-\tfrac{1}{2}\alpha-\tfrac{1}{2})x^2 + (\tfrac{827}{2}\alpha-\tfrac{1159}{2})x + (-11829\alpha-6556)
\end{align*}
$$
Reducing the coefficients modulo v1 gets tells us that $\overline{E}$ and $\overline{E'}$ are both ordinary. So it doesn't seem like .minimal_model() is at fault here.

Best Answer

If you take a curve $y^2 = x^3 + a x + b$ with relatively small $a$ and $b$, you are preselecting for curves which are already in minimal Weierstrass form at (a prime above) $3$, and thus which will either have bad reduction or supersingular reduction at $3$. So your sample space is biasing the results. In particular, I think that the only way your curve could have good reduction at (all primes above) $3$ is if $27|a$ and $27|b$. This is already outside your range if I understand it.

As a comparison, if you look at $y^2 = x^3 + a x + b$ but now with $a,b \in \mathbf{Z}$ in a small box you won't find any curves with good ordinary reduction at the prime $3$ either.

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