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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 129360.s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.s1 | 129360h6 | \([0, -1, 0, -100614656, -388420808544]\) | \(467508233804095622882/315748125\) | \(76077979971840000\) | \([2]\) | \(9437184\) | \(2.9890\) | |
| 129360.s2 | 129360h4 | \([0, -1, 0, -6289656, -6064988544]\) | \(228410605013945764/187597265625\) | \(22600325840400000000\) | \([2, 2]\) | \(4718592\) | \(2.6425\) | |
| 129360.s3 | 129360h5 | \([0, -1, 0, -4931376, -8758729440]\) | \(-55043996611705922/105743408203125\) | \(-25478361562500000000000\) | \([2]\) | \(9437184\) | \(2.9890\) | |
| 129360.s4 | 129360h3 | \([0, -1, 0, -4080736, 3139446640]\) | \(62380825826921284/787768887675\) | \(94904547190861900800\) | \([2]\) | \(4718592\) | \(2.6425\) | |
| 129360.s5 | 129360h2 | \([0, -1, 0, -479236, -50041760]\) | \(404151985581136/197735855625\) | \(5955436973676960000\) | \([2, 2]\) | \(2359296\) | \(2.2959\) | |
| 129360.s6 | 129360h1 | \([0, -1, 0, 109009, -6041034]\) | \(76102438406144/52315569075\) | \(-98477990177674800\) | \([2]\) | \(1179648\) | \(1.9493\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 129360.s have rank \(0\).
Complex multiplication
The elliptic curves in class 129360.s do not have complex multiplication.Modular form 129360.2.a.s
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
