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SageMath
E = EllipticCurve("gg1")
E.isogeny_class()
Elliptic curves in class 193200gg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.fo1 | 193200gg1 | \([0, 1, 0, -11908, -503812]\) | \(46689225424/36225\) | \(144900000000\) | \([2]\) | \(393216\) | \(1.0731\) | \(\Gamma_0(N)\)-optimal |
| 193200.fo2 | 193200gg2 | \([0, 1, 0, -9408, -718812]\) | \(-5756278756/10498005\) | \(-167968080000000\) | \([2]\) | \(786432\) | \(1.4197\) |
Rank
sage: E.rank()
The elliptic curves in class 193200gg have rank \(1\).
Complex multiplication
The elliptic curves in class 193200gg do not have complex multiplication.Modular form 193200.2.a.gg
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 Cremona 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 Cremona labels.
