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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 690.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 690.e1 | 690e3 | \([1, 0, 1, -10644, 420826]\) | \(133345896593725369/340006815000\) | \(340006815000\) | \([2]\) | \(1920\) | \(1.0891\) | |
| 690.e2 | 690e2 | \([1, 0, 1, -924, 922]\) | \(87109155423289/49979073600\) | \(49979073600\) | \([2, 2]\) | \(960\) | \(0.74257\) | |
| 690.e3 | 690e1 | \([1, 0, 1, -604, -5734]\) | \(24310870577209/114462720\) | \(114462720\) | \([2]\) | \(480\) | \(0.39599\) | \(\Gamma_0(N)\)-optimal |
| 690.e4 | 690e4 | \([1, 0, 1, 3676, 8282]\) | \(5495662324535111/3207841648920\) | \(-3207841648920\) | \([2]\) | \(1920\) | \(1.0891\) |
Rank
sage: E.rank()
The elliptic curves in class 690.e have rank \(1\).
Complex multiplication
The elliptic curves in class 690.e do not have complex multiplication.Modular form 690.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
