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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 167310l
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 167310.fg6 | 167310l1 | \([1, -1, 1, 387823, -8931711]\) | \(1833318007919/1070530560\) | \(-3766922728959836160\) | \([2]\) | \(2949120\) | \(2.2538\) | \(\Gamma_0(N)\)-optimal |
| 167310.fg5 | 167310l2 | \([1, -1, 1, -1559057, -70453119]\) | \(119102750067601/68309049600\) | \(240362042099839545600\) | \([2, 2]\) | \(5898240\) | \(2.6004\) | |
| 167310.fg3 | 167310l3 | \([1, -1, 1, -16282337, 25188806049]\) | \(135670761487282321/643043610000\) | \(2262705690738417210000\) | \([2, 2]\) | \(11796480\) | \(2.9470\) | |
| 167310.fg2 | 167310l4 | \([1, -1, 1, -17985857, -29290444959]\) | \(182864522286982801/463015182960\) | \(1629231786288773512560\) | \([2]\) | \(11796480\) | \(2.9470\) | |
| 167310.fg1 | 167310l5 | \([1, -1, 1, -260220317, 1615762010841]\) | \(553808571467029327441/12529687500\) | \(44088759717904687500\) | \([2]\) | \(23592960\) | \(3.2935\) | |
| 167310.fg4 | 167310l6 | \([1, -1, 1, -7916837, 51031508649]\) | \(-15595206456730321/310672490129100\) | \(-1093176886356104709545100\) | \([2]\) | \(23592960\) | \(3.2935\) |
Rank
sage: E.rank()
The elliptic curves in class 167310l have rank \(1\).
Complex multiplication
The elliptic curves in class 167310l do not have complex multiplication.Modular form 167310.2.a.l
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.
