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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 5040.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5040.c1 | 5040be1 | \([0, 0, 0, -5088, 139687]\) | \(1248870793216/42525\) | \(496011600\) | \([2]\) | \(3840\) | \(0.76020\) | \(\Gamma_0(N)\)-optimal |
5040.c2 | 5040be2 | \([0, 0, 0, -4863, 152602]\) | \(-68150496976/14467005\) | \(-2699890341120\) | \([2]\) | \(7680\) | \(1.1068\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.c have rank \(1\).
Complex multiplication
The elliptic curves in class 5040.c do not have complex multiplication.Modular form 5040.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.