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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 690.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 690.f1 | 690f3 | \([1, 0, 1, -1473, -21872]\) | \(353108405631241/172500\) | \(172500\) | \([2]\) | \(256\) | \(0.33992\) | |
| 690.f2 | 690f2 | \([1, 0, 1, -93, -344]\) | \(87587538121/1904400\) | \(1904400\) | \([2, 2]\) | \(128\) | \(-0.0066500\) | |
| 690.f3 | 690f1 | \([1, 0, 1, -13, 8]\) | \(217081801/88320\) | \(88320\) | \([2]\) | \(64\) | \(-0.35322\) | \(\Gamma_0(N)\)-optimal |
| 690.f4 | 690f4 | \([1, 0, 1, 7, -1024]\) | \(46268279/453342420\) | \(-453342420\) | \([4]\) | \(256\) | \(0.33992\) |
Rank
sage: E.rank()
The elliptic curves in class 690.f have rank \(0\).
Complex multiplication
The elliptic curves in class 690.f do not have complex multiplication.Modular form 690.2.a.f
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.
