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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 422370.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
422370.c1 | 422370c2 | \([1, -1, 0, -404671140, -3659509626800]\) | \(-1639709351099641/345558220800\) | \(-1544488813438765579055923200\) | \([]\) | \(388862208\) | \(3.9386\) | |
422370.c2 | 422370c1 | \([1, -1, 0, 35162235, 28669155925]\) | \(1075696074359/702000000\) | \(-3137622205959724158000000\) | \([]\) | \(129620736\) | \(3.3893\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 422370.c have rank \(1\).
Complex multiplication
The elliptic curves in class 422370.c do not have complex multiplication.Modular form 422370.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.