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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 74382x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 74382.bg6 | 74382x1 | \([1, 1, 1, 5106583, -568213801]\) | \(125177609053596564863/73635189229502208\) | \(-8663106377661705268992\) | \([2]\) | \(4915200\) | \(2.8988\) | \(\Gamma_0(N)\)-optimal |
| 74382.bg5 | 74382x2 | \([1, 1, 1, -20612537, -4590684169]\) | \(8232463578739844255617/4687062591766850064\) | \(551428226858778143179536\) | \([2, 2]\) | \(9830400\) | \(3.2454\) | |
| 74382.bg3 | 74382x3 | \([1, 1, 1, -211203917, 1175932323551]\) | \(8856076866003496152467137/46664863048067576004\) | \(5490074472742102249294596\) | \([2, 2]\) | \(19660800\) | \(3.5920\) | |
| 74382.bg2 | 74382x4 | \([1, 1, 1, -241527077, -1442037413041]\) | \(13244420128496241770842177/29965867631164664892\) | \(3525454360938891659878908\) | \([2]\) | \(19660800\) | \(3.5920\) | |
| 74382.bg4 | 74382x5 | \([1, 1, 1, -96892307, 2444013875603]\) | \(-855073332201294509246497/21439133060285771735058\) | \(-2522292565409560758857838642\) | \([2]\) | \(39321600\) | \(3.9385\) | |
| 74382.bg1 | 74382x6 | \([1, 1, 1, -3374977607, 75465135093179]\) | \(36136672427711016379227705697/1011258101510224722\) | \(118973504384576428318578\) | \([2]\) | \(39321600\) | \(3.9385\) |
Rank
sage: E.rank()
The elliptic curves in class 74382x have rank \(1\).
Complex multiplication
The elliptic curves in class 74382x do not have complex multiplication.Modular form 74382.2.a.x
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
