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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 8925b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8925.u4 | 8925b1 | \([1, 1, 0, -97625, 11700000]\) | \(6585576176607121/187425\) | \(2928515625\) | \([2]\) | \(24576\) | \(1.3262\) | \(\Gamma_0(N)\)-optimal |
| 8925.u3 | 8925b2 | \([1, 1, 0, -97750, 11668375]\) | \(6610905152742241/35128130625\) | \(548877041015625\) | \([2, 2]\) | \(49152\) | \(1.6728\) | |
| 8925.u2 | 8925b3 | \([1, 1, 0, -152875, -3050000]\) | \(25288177725059761/14387797265625\) | \(224809332275390625\) | \([2, 2]\) | \(98304\) | \(2.0194\) | |
| 8925.u5 | 8925b4 | \([1, 1, 0, -44625, 24365250]\) | \(-629004249876241/16074715228425\) | \(-251167425444140625\) | \([2]\) | \(98304\) | \(2.0194\) | |
| 8925.u1 | 8925b5 | \([1, 1, 0, -1793500, -923440625]\) | \(40832710302042509761/91556816413125\) | \(1430575256455078125\) | \([2]\) | \(196608\) | \(2.3659\) | |
| 8925.u6 | 8925b6 | \([1, 1, 0, 605750, -23532875]\) | \(1573196002879828319/926055908203125\) | \(-14469623565673828125\) | \([2]\) | \(196608\) | \(2.3659\) |
Rank
sage: E.rank()
The elliptic curves in class 8925b have rank \(1\).
Complex multiplication
The elliptic curves in class 8925b do not have complex multiplication.Modular form 8925.2.a.b
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.
