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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 255024.ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 255024.ec1 | 255024ec6 | \([0, 0, 0, -9918301539, -380192954591518]\) | \(36136672427711016379227705697/1011258101510224722\) | \(3019600510979906856296448\) | \([2]\) | \(157286400\) | \(4.2080\) | |
| 255024.ec2 | 255024ec3 | \([0, 0, 0, -709793859, 7264334548802]\) | \(13244420128496241770842177/29965867631164664892\) | \(89477601292775590732873728\) | \([2]\) | \(78643200\) | \(3.8615\) | |
| 255024.ec3 | 255024ec4 | \([0, 0, 0, -620680899, -5924673824830]\) | \(8856076866003496152467137/46664863048067576004\) | \(139340534423721012866727936\) | \([2, 2]\) | \(78643200\) | \(3.8615\) | |
| 255024.ec4 | 255024ec5 | \([0, 0, 0, -284744739, -12312903030622]\) | \(-855073332201294509246497/21439133060285771735058\) | \(-64016908291884349828535427072\) | \([2]\) | \(157286400\) | \(4.2080\) | |
| 255024.ec5 | 255024ec2 | \([0, 0, 0, -60575619, 23084143490]\) | \(8232463578739844255617/4687062591766850064\) | \(13995493906014346021502976\) | \([2, 2]\) | \(39321600\) | \(3.5149\) | |
| 255024.ec6 | 255024ec1 | \([0, 0, 0, 15007101, 2873324162]\) | \(125177609053596564863/73635189229502208\) | \(-219873496876265921052672\) | \([2]\) | \(19660800\) | \(3.1683\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 255024.ec have rank \(1\).
Complex multiplication
The elliptic curves in class 255024.ec do not have complex multiplication.Modular form 255024.2.a.ec
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
