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SageMath
E = EllipticCurve("fc1")
E.isogeny_class()
Elliptic curves in class 248640fc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 248640.fc4 | 248640fc1 | \([0, 1, 0, 5119, 116319]\) | \(56578878719/54390000\) | \(-14258012160000\) | \([2]\) | \(491520\) | \(1.2111\) | \(\Gamma_0(N)\)-optimal |
| 248640.fc3 | 248640fc2 | \([0, 1, 0, -26881, 1031519]\) | \(8194759433281/2958272100\) | \(775493281382400\) | \([2, 2]\) | \(983040\) | \(1.5577\) | |
| 248640.fc1 | 248640fc3 | \([0, 1, 0, -382081, 90755039]\) | \(23531588875176481/6398929110\) | \(1677440872611840\) | \([2]\) | \(1966080\) | \(1.9043\) | |
| 248640.fc2 | 248640fc4 | \([0, 1, 0, -183681, -29607201]\) | \(2614441086442081/74385450090\) | \(19499699428392960\) | \([2]\) | \(1966080\) | \(1.9043\) |
Rank
sage: E.rank()
The elliptic curves in class 248640fc have rank \(0\).
Complex multiplication
The elliptic curves in class 248640fc do not have complex multiplication.Modular form 248640.2.a.fc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
