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SageMath
E = EllipticCurve("gv1")
E.isogeny_class()
Elliptic curves in class 297024gv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 297024.gv4 | 297024gv1 | \([0, 1, 0, -105217, 13101407]\) | \(491411892194497/78897\) | \(20682375168\) | \([2]\) | \(786432\) | \(1.3819\) | \(\Gamma_0(N)\)-optimal |
| 297024.gv3 | 297024gv2 | \([0, 1, 0, -105537, 13017375]\) | \(495909170514577/6224736609\) | \(1631777353629696\) | \([2, 2]\) | \(1572864\) | \(1.7285\) | |
| 297024.gv5 | 297024gv3 | \([0, 1, 0, -18177, 34001247]\) | \(-2533811507137/1904381781393\) | \(-499222257701486592\) | \([2]\) | \(3145728\) | \(2.0750\) | |
| 297024.gv2 | 297024gv4 | \([0, 1, 0, -198017, -13339425]\) | \(3275619238041697/1605271262049\) | \(420812229718573056\) | \([2, 2]\) | \(3145728\) | \(2.0750\) | |
| 297024.gv6 | 297024gv5 | \([0, 1, 0, 721343, -101414113]\) | \(158346567380527343/108665074944153\) | \(-28485897406160044032\) | \([2]\) | \(6291456\) | \(2.4216\) | |
| 297024.gv1 | 297024gv6 | \([0, 1, 0, -2597057, -1610620257]\) | \(7389727131216686257/6115533215337\) | \(1603150339201302528\) | \([2]\) | \(6291456\) | \(2.4216\) |
Rank
sage: E.rank()
The elliptic curves in class 297024gv have rank \(1\).
Complex multiplication
The elliptic curves in class 297024gv do not have complex multiplication.Modular form 297024.2.a.gv
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
