Show commands:
SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 18240.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18240.bd1 | 18240cb3 | \([0, -1, 0, -39745, 3058657]\) | \(26487576322129/44531250\) | \(11673600000000\) | \([4]\) | \(49152\) | \(1.4026\) | |
| 18240.bd2 | 18240cb2 | \([0, -1, 0, -3265, 16225]\) | \(14688124849/8122500\) | \(2129264640000\) | \([2, 2]\) | \(24576\) | \(1.0561\) | |
| 18240.bd3 | 18240cb1 | \([0, -1, 0, -1985, -33183]\) | \(3301293169/22800\) | \(5976883200\) | \([2]\) | \(12288\) | \(0.70949\) | \(\Gamma_0(N)\)-optimal |
| 18240.bd4 | 18240cb4 | \([0, -1, 0, 12735, 115425]\) | \(871257511151/527800050\) | \(-138359616307200\) | \([2]\) | \(49152\) | \(1.4026\) |
Rank
sage: E.rank()
The elliptic curves in class 18240.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 18240.bd do not have complex multiplication.Modular form 18240.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
