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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 2420.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2420.e1 | 2420d2 | \([0, 1, 0, -52796, -4688620]\) | \(-296587984/125\) | \(-6859484192000\) | \([]\) | \(4752\) | \(1.4242\) | |
2420.e2 | 2420d1 | \([0, 1, 0, 444, -24796]\) | \(176/5\) | \(-274379367680\) | \([3]\) | \(1584\) | \(0.87489\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2420.e have rank \(1\).
Complex multiplication
The elliptic curves in class 2420.e do not have complex multiplication.Modular form 2420.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.