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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 80688.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 80688.e1 | 80688e6 | \([0, -1, 0, -646064, 200091360]\) | \(3065617154/9\) | \(87553921370112\) | \([2]\) | \(552960\) | \(1.9046\) | |
| 80688.e2 | 80688e4 | \([0, -1, 0, -108144, -13651152]\) | \(28756228/3\) | \(14592320228352\) | \([2]\) | \(276480\) | \(1.5580\) | |
| 80688.e3 | 80688e3 | \([0, -1, 0, -40904, 3051264]\) | \(1556068/81\) | \(393992646165504\) | \([2, 2]\) | \(276480\) | \(1.5580\) | |
| 80688.e4 | 80688e2 | \([0, -1, 0, -7284, -176256]\) | \(35152/9\) | \(10944240171264\) | \([2, 2]\) | \(138240\) | \(1.2114\) | |
| 80688.e5 | 80688e1 | \([0, -1, 0, 1121, -18242]\) | \(2048/3\) | \(-228005003568\) | \([2]\) | \(69120\) | \(0.86486\) | \(\Gamma_0(N)\)-optimal |
| 80688.e6 | 80688e5 | \([0, -1, 0, 26336, 12034528]\) | \(207646/6561\) | \(-63826808678811648\) | \([2]\) | \(552960\) | \(1.9046\) |
Rank
sage: E.rank()
The elliptic curves in class 80688.e have rank \(1\).
Complex multiplication
The elliptic curves in class 80688.e do not have complex multiplication.Modular form 80688.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
