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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 34200bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34200.j3 | 34200bg1 | \([0, 0, 0, -57450, -2413375]\) | \(115060504576/52780005\) | \(9619155911250000\) | \([2]\) | \(245760\) | \(1.7617\) | \(\Gamma_0(N)\)-optimal |
| 34200.j2 | 34200bg2 | \([0, 0, 0, -463575, 119830250]\) | \(3778298043856/59213025\) | \(172665180900000000\) | \([2, 2]\) | \(491520\) | \(2.1083\) | |
| 34200.j4 | 34200bg3 | \([0, 0, 0, -36075, 332297750]\) | \(-445138564/4089438495\) | \(-47699210605680000000\) | \([2]\) | \(983040\) | \(2.4549\) | |
| 34200.j1 | 34200bg4 | \([0, 0, 0, -7389075, 7730954750]\) | \(3825131988299044/961875\) | \(11219310000000000\) | \([2]\) | \(983040\) | \(2.4549\) |
Rank
sage: E.rank()
The elliptic curves in class 34200bg have rank \(1\).
Complex multiplication
The elliptic curves in class 34200bg do not have complex multiplication.Modular form 34200.2.a.bg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
