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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 145475.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 145475.f1 | 145475f4 | \([1, -1, 1, -782755, -266290128]\) | \(22930509321/6875\) | \(15902292763671875\) | \([2]\) | \(1182720\) | \(2.0868\) | |
| 145475.f2 | 145475f3 | \([1, -1, 1, -386005, 90255872]\) | \(2749884201/73205\) | \(169327613347578125\) | \([2]\) | \(1182720\) | \(2.0868\) | |
| 145475.f3 | 145475f2 | \([1, -1, 1, -55380, -2980378]\) | \(8120601/3025\) | \(6997008816015625\) | \([2, 2]\) | \(591360\) | \(1.7402\) | |
| 145475.f4 | 145475f1 | \([1, -1, 1, 10745, -335378]\) | \(59319/55\) | \(-127218342109375\) | \([2]\) | \(295680\) | \(1.3937\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 145475.f have rank \(1\).
Complex multiplication
The elliptic curves in class 145475.f do not have complex multiplication.Modular form 145475.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
