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SageMath
E = EllipticCurve("pd1")
E.isogeny_class()
Elliptic curves in class 141120.pd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141120.pd1 | 141120ce4 | \([0, 0, 0, -99372, -11947376]\) | \(38614472/405\) | \(1138205622435840\) | \([2]\) | \(786432\) | \(1.7056\) | |
| 141120.pd2 | 141120ce2 | \([0, 0, 0, -11172, 153664]\) | \(438976/225\) | \(79042057113600\) | \([2, 2]\) | \(393216\) | \(1.3590\) | |
| 141120.pd3 | 141120ce1 | \([0, 0, 0, -8967, 326536]\) | \(14526784/15\) | \(82335476160\) | \([2]\) | \(196608\) | \(1.0124\) | \(\Gamma_0(N)\)-optimal |
| 141120.pd4 | 141120ce3 | \([0, 0, 0, 41748, 1190896]\) | \(2863288/1875\) | \(-5269470474240000\) | \([2]\) | \(786432\) | \(1.7056\) |
Rank
sage: E.rank()
The elliptic curves in class 141120.pd have rank \(1\).
Complex multiplication
The elliptic curves in class 141120.pd do not have complex multiplication.Modular form 141120.2.a.pd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
