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SageMath
E = EllipticCurve("gh1")
E.isogeny_class()
Elliptic curves in class 193200.gh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
193200.gh1 | 193200bc2 | \([0, 1, 0, -169008, -14964012]\) | \(8341959848041/3327411150\) | \(212954313600000000\) | \([2]\) | \(2211840\) | \(2.0232\) | |
193200.gh2 | 193200bc1 | \([0, 1, 0, -77008, 8035988]\) | \(789145184521/17996580\) | \(1151781120000000\) | \([2]\) | \(1105920\) | \(1.6766\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 193200.gh have rank \(1\).
Complex multiplication
The elliptic curves in class 193200.gh do not have complex multiplication.Modular form 193200.2.a.gh
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.