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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 37845.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37845.d1 | 37845d8 | \([1, -1, 1, -16349198, 25448520512]\) | \(1114544804970241/405\) | \(175618611408645\) | \([2]\) | \(802816\) | \(2.5238\) | |
| 37845.d2 | 37845d6 | \([1, -1, 1, -1021973, 397703972]\) | \(272223782641/164025\) | \(71125537620501225\) | \([2, 2]\) | \(401408\) | \(2.1772\) | |
| 37845.d3 | 37845d7 | \([1, -1, 1, -832748, 549386732]\) | \(-147281603041/215233605\) | \(-93330930465621707445\) | \([2]\) | \(802816\) | \(2.5238\) | |
| 37845.d4 | 37845d4 | \([1, -1, 1, -605678, -181279114]\) | \(56667352321/15\) | \(6504393015135\) | \([2]\) | \(200704\) | \(1.8307\) | |
| 37845.d5 | 37845d3 | \([1, -1, 1, -75848, 3737522]\) | \(111284641/50625\) | \(21952326426080625\) | \([2, 2]\) | \(200704\) | \(1.8307\) | |
| 37845.d6 | 37845d2 | \([1, -1, 1, -38003, -2802094]\) | \(13997521/225\) | \(97565895227025\) | \([2, 2]\) | \(100352\) | \(1.4841\) | |
| 37845.d7 | 37845d1 | \([1, -1, 1, -158, -122668]\) | \(-1/15\) | \(-6504393015135\) | \([2]\) | \(50176\) | \(1.1375\) | \(\Gamma_0(N)\)-optimal |
| 37845.d8 | 37845d5 | \([1, -1, 1, 264757, 27852356]\) | \(4733169839/3515625\) | \(-1524467112922265625\) | \([2]\) | \(401408\) | \(2.1772\) |
Rank
sage: E.rank()
The elliptic curves in class 37845.d have rank \(0\).
Complex multiplication
The elliptic curves in class 37845.d do not have complex multiplication.Modular form 37845.2.a.d
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
