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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 2640.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2640.f1 | 2640p4 | \([0, -1, 0, -1556, 13356]\) | \(1628514404944/664335375\) | \(170069856000\) | \([2]\) | \(3456\) | \(0.85261\) | |
| 2640.f2 | 2640p2 | \([0, -1, 0, -716, -7140]\) | \(158792223184/16335\) | \(4181760\) | \([2]\) | \(1152\) | \(0.30330\) | |
| 2640.f3 | 2640p1 | \([0, -1, 0, -41, -120]\) | \(-488095744/200475\) | \(-3207600\) | \([2]\) | \(576\) | \(-0.043272\) | \(\Gamma_0(N)\)-optimal |
| 2640.f4 | 2640p3 | \([0, -1, 0, 319, 1356]\) | \(223673040896/187171875\) | \(-2994750000\) | \([2]\) | \(1728\) | \(0.50603\) |
Rank
sage: E.rank()
The elliptic curves in class 2640.f have rank \(1\).
Complex multiplication
The elliptic curves in class 2640.f do not have complex multiplication.Modular form 2640.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
