Show commands:
SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 11550.bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11550.bt1 | 11550bq4 | \([1, 1, 1, -6652813, 6601973531]\) | \(2084105208962185000201/31185000\) | \(487265625000\) | \([2]\) | \(294912\) | \(2.2462\) | |
11550.bt2 | 11550bq3 | \([1, 1, 1, -450813, 84609531]\) | \(648474704552553481/176469171805080\) | \(2757330809454375000\) | \([2]\) | \(294912\) | \(2.2462\) | |
11550.bt3 | 11550bq2 | \([1, 1, 1, -415813, 103019531]\) | \(508859562767519881/62240270400\) | \(972504225000000\) | \([2, 2]\) | \(147456\) | \(1.8997\) | |
11550.bt4 | 11550bq1 | \([1, 1, 1, -23813, 1883531]\) | \(-95575628340361/43812679680\) | \(-684573120000000\) | \([4]\) | \(73728\) | \(1.5531\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 11550.bt have rank \(1\).
Complex multiplication
The elliptic curves in class 11550.bt do not have complex multiplication.Modular form 11550.2.a.bt
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.