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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 286650z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.z4 | 286650z1 | \([1, -1, 0, -5330817, -491358659]\) | \(12501706118329/7156531200\) | \(9590436263116800000000\) | \([2]\) | \(23592960\) | \(2.9076\) | \(\Gamma_0(N)\)-optimal |
| 286650.z2 | 286650z2 | \([1, -1, 0, -61778817, -186487518659]\) | \(19458380202497209/47698560000\) | \(63920632320090000000000\) | \([2, 2]\) | \(47185920\) | \(3.2542\) | |
| 286650.z3 | 286650z3 | \([1, -1, 0, -38846817, -326670834659]\) | \(-4837870546133689/31603162500000\) | \(-42351260296213476562500000\) | \([2]\) | \(94371840\) | \(3.6008\) | |
| 286650.z1 | 286650z4 | \([1, -1, 0, -987878817, -11950735818659]\) | \(79560762543506753209/479824800\) | \(643011122743762500000\) | \([2]\) | \(94371840\) | \(3.6008\) |
Rank
sage: E.rank()
The elliptic curves in class 286650z have rank \(1\).
Complex multiplication
The elliptic curves in class 286650z do not have complex multiplication.Modular form 286650.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
