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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1216.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1216.e1 | 1216b3 | \([0, -1, 0, -5473, -1251871]\) | \(-69173457625/2550136832\) | \(-668503069687808\) | \([]\) | \(3456\) | \(1.5249\) | |
| 1216.e2 | 1216b1 | \([0, -1, 0, -993, 12385]\) | \(-413493625/152\) | \(-39845888\) | \([]\) | \(384\) | \(0.42628\) | \(\Gamma_0(N)\)-optimal |
| 1216.e3 | 1216b2 | \([0, -1, 0, 607, 45601]\) | \(94196375/3511808\) | \(-920599396352\) | \([]\) | \(1152\) | \(0.97558\) |
Rank
sage: E.rank()
The elliptic curves in class 1216.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1216.e do not have complex multiplication.Modular form 1216.2.a.e
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 LMFDB numbering.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
