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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 10780.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10780.a1 | 10780g2 | \([0, 1, 0, -6876, 43924]\) | \(1193895376/660275\) | \(19886257529600\) | \([2]\) | \(27648\) | \(1.2423\) | |
10780.a2 | 10780g1 | \([0, 1, 0, -4181, -104840]\) | \(4294967296/29645\) | \(55803273680\) | \([2]\) | \(13824\) | \(0.89568\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10780.a have rank \(1\).
Complex multiplication
The elliptic curves in class 10780.a do not have complex multiplication.Modular form 10780.2.a.a
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.