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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 6069.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6069.b1 | 6069b5 | \([1, 1, 1, -226582, -41607616]\) | \(53297461115137/147\) | \(3548222643\) | \([2]\) | \(18432\) | \(1.4908\) | |
| 6069.b2 | 6069b3 | \([1, 1, 1, -14167, -654004]\) | \(13027640977/21609\) | \(521588728521\) | \([2, 2]\) | \(9216\) | \(1.1442\) | |
| 6069.b3 | 6069b4 | \([1, 1, 1, -11277, 453444]\) | \(6570725617/45927\) | \(1108566131463\) | \([2]\) | \(9216\) | \(1.1442\) | |
| 6069.b4 | 6069b6 | \([1, 1, 1, -9832, -1056292]\) | \(-4354703137/17294403\) | \(-417444845726307\) | \([2]\) | \(18432\) | \(1.4908\) | |
| 6069.b5 | 6069b2 | \([1, 1, 1, -1162, -3754]\) | \(7189057/3969\) | \(95802011361\) | \([2, 2]\) | \(4608\) | \(0.79766\) | |
| 6069.b6 | 6069b1 | \([1, 1, 1, 283, -286]\) | \(103823/63\) | \(-1520666847\) | \([2]\) | \(2304\) | \(0.45109\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6069.b have rank \(0\).
Complex multiplication
The elliptic curves in class 6069.b do not have complex multiplication.Modular form 6069.2.a.b
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
