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SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 123840eq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123840.u2 | 123840eq1 | \([0, 0, 0, -49548, 3801872]\) | \(70393838689/8062500\) | \(1540767744000000\) | \([2]\) | \(442368\) | \(1.6465\) | \(\Gamma_0(N)\)-optimal |
| 123840.u1 | 123840eq2 | \([0, 0, 0, -769548, 259833872]\) | \(263732349218689/4160250\) | \(795036155904000\) | \([2]\) | \(884736\) | \(1.9930\) |
Rank
sage: E.rank()
The elliptic curves in class 123840eq have rank \(1\).
Complex multiplication
The elliptic curves in class 123840eq do not have complex multiplication.Modular form 123840.2.a.eq
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
