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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 5355f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5355.q4 | 5355f1 | \([1, -1, 0, -35145, -2527200]\) | \(6585576176607121/187425\) | \(136632825\) | \([2]\) | \(8192\) | \(1.0708\) | \(\Gamma_0(N)\)-optimal |
| 5355.q3 | 5355f2 | \([1, -1, 0, -35190, -2520369]\) | \(6610905152742241/35128130625\) | \(25608407225625\) | \([2, 2]\) | \(16384\) | \(1.4174\) | |
| 5355.q2 | 5355f3 | \([1, -1, 0, -55035, 658800]\) | \(25288177725059761/14387797265625\) | \(10488704206640625\) | \([2, 2]\) | \(32768\) | \(1.7639\) | |
| 5355.q5 | 5355f4 | \([1, -1, 0, -16065, -5262894]\) | \(-629004249876241/16074715228425\) | \(-11718467401521825\) | \([2]\) | \(32768\) | \(1.7639\) | |
| 5355.q1 | 5355f5 | \([1, -1, 0, -645660, 199463175]\) | \(40832710302042509761/91556816413125\) | \(66744919165168125\) | \([2]\) | \(65536\) | \(2.1105\) | |
| 5355.q6 | 5355f6 | \([1, -1, 0, 218070, 5083101]\) | \(1573196002879828319/926055908203125\) | \(-675094757080078125\) | \([2]\) | \(65536\) | \(2.1105\) |
Rank
sage: E.rank()
The elliptic curves in class 5355f have rank \(1\).
Complex multiplication
The elliptic curves in class 5355f do not have complex multiplication.Modular form 5355.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
