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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 62475w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 62475.cg3 | 62475w1 | \([1, 1, 0, -410400, 101023875]\) | \(4158523459441/16065\) | \(29531737265625\) | \([2]\) | \(442368\) | \(1.7982\) | \(\Gamma_0(N)\)-optimal |
| 62475.cg2 | 62475w2 | \([1, 1, 0, -416525, 97845000]\) | \(4347507044161/258084225\) | \(474427359172265625\) | \([2, 2]\) | \(884736\) | \(2.1447\) | |
| 62475.cg4 | 62475w3 | \([1, 1, 0, 312350, 404701375]\) | \(1833318007919/39525924375\) | \(-72659148074912109375\) | \([2]\) | \(1769472\) | \(2.4913\) | |
| 62475.cg1 | 62475w4 | \([1, 1, 0, -1243400, -412336875]\) | \(115650783909361/27072079335\) | \(49765672838803359375\) | \([2]\) | \(1769472\) | \(2.4913\) |
Rank
sage: E.rank()
The elliptic curves in class 62475w have rank \(1\).
Complex multiplication
The elliptic curves in class 62475w do not have complex multiplication.Modular form 62475.2.a.w
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.
