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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 4032.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4032.bd1 | 4032bc3 | \([0, 0, 0, -4044, 92752]\) | \(306182024/21609\) | \(516193026048\) | \([2]\) | \(4096\) | \(0.99483\) | |
| 4032.bd2 | 4032bc2 | \([0, 0, 0, -804, -7040]\) | \(19248832/3969\) | \(11851370496\) | \([2, 2]\) | \(2048\) | \(0.64825\) | |
| 4032.bd3 | 4032bc1 | \([0, 0, 0, -759, -8048]\) | \(1036433728/63\) | \(2939328\) | \([2]\) | \(1024\) | \(0.30168\) | \(\Gamma_0(N)\)-optimal |
| 4032.bd4 | 4032bc4 | \([0, 0, 0, 1716, -42320]\) | \(23393656/45927\) | \(-1097098297344\) | \([2]\) | \(4096\) | \(0.99483\) |
Rank
sage: E.rank()
The elliptic curves in class 4032.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 4032.bd do not have complex multiplication.Modular form 4032.2.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.
