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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 10350.bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10350.bg1 | 10350bj3 | \([1, -1, 1, -24840005, -47645185503]\) | \(148809678420065817601/20700\) | \(235785937500\) | \([2]\) | \(294912\) | \(2.5070\) | |
| 10350.bg2 | 10350bj5 | \([1, -1, 1, -5811755, 4620806997]\) | \(1905890658841300321/293666194803750\) | \(3345041500186464843750\) | \([2]\) | \(589824\) | \(2.8536\) | |
| 10350.bg3 | 10350bj4 | \([1, -1, 1, -1593005, -703255503]\) | \(39248884582600321/3935264062500\) | \(44825117211914062500\) | \([2, 2]\) | \(294912\) | \(2.5070\) | |
| 10350.bg4 | 10350bj2 | \([1, -1, 1, -1552505, -744160503]\) | \(36330796409313601/428490000\) | \(4880768906250000\) | \([2, 2]\) | \(147456\) | \(2.1605\) | |
| 10350.bg5 | 10350bj1 | \([1, -1, 1, -94505, -12244503]\) | \(-8194759433281/965779200\) | \(-11000828700000000\) | \([2]\) | \(73728\) | \(1.8139\) | \(\Gamma_0(N)\)-optimal |
| 10350.bg6 | 10350bj6 | \([1, -1, 1, 1977745, -3409884003]\) | \(75108181893694559/484313964843750\) | \(-5516638755798339843750\) | \([2]\) | \(589824\) | \(2.8536\) |
Rank
sage: E.rank()
The elliptic curves in class 10350.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 10350.bg do not have complex multiplication.Modular form 10350.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
