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SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 141120.ei
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141120.ei1 | 141120mb4 | \([0, 0, 0, -24908268, -47847901808]\) | \(608119035935048/826875\) | \(2323836479139840000\) | \([2]\) | \(4718592\) | \(2.7984\) | |
| 141120.ei2 | 141120mb3 | \([0, 0, 0, -3951948, 2020138288]\) | \(2428799546888/778248135\) | \(2187176303474584289280\) | \([2]\) | \(4718592\) | \(2.7984\) | |
| 141120.ei3 | 141120mb2 | \([0, 0, 0, -1570548, -733712672]\) | \(1219555693504/43758225\) | \(15372178309510041600\) | \([2, 2]\) | \(2359296\) | \(2.4518\) | |
| 141120.ei4 | 141120mb1 | \([0, 0, 0, 36897, -40582388]\) | \(1012048064/130203045\) | \(-714688647170460480\) | \([2]\) | \(1179648\) | \(2.1052\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 141120.ei have rank \(1\).
Complex multiplication
The elliptic curves in class 141120.ei do not have complex multiplication.Modular form 141120.2.a.ei
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.
