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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 51714a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51714.i2 | 51714a1 | \([1, -1, 0, -31212, -1057536]\) | \(955671625/413712\) | \(1455746518850832\) | \([2]\) | \(172032\) | \(1.6057\) | \(\Gamma_0(N)\)-optimal |
| 51714.i1 | 51714a2 | \([1, -1, 0, -426672, -107119908]\) | \(2441288319625/1217268\) | \(4283254180464948\) | \([2]\) | \(344064\) | \(1.9523\) |
Rank
sage: E.rank()
The elliptic curves in class 51714a have rank \(0\).
Complex multiplication
The elliptic curves in class 51714a do not have complex multiplication.Modular form 51714.2.a.a
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
