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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 331056k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 331056.k1 | 331056k1 | \([0, 0, 0, -360459, 83435066]\) | \(-26436959739/50578\) | \(-9909265739636736\) | \([]\) | \(4147200\) | \(1.9589\) | \(\Gamma_0(N)\)-optimal |
| 331056.k2 | 331056k2 | \([0, 0, 0, 597861, 409753674]\) | \(165469149/603592\) | \(-86208646460643311616\) | \([]\) | \(12441600\) | \(2.5082\) |
Rank
sage: E.rank()
The elliptic curves in class 331056k have rank \(2\).
Complex multiplication
The elliptic curves in class 331056k do not have complex multiplication.Modular form 331056.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
