Show commands:
SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 139638.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139638.bd1 | 139638x1 | \([1, 1, 1, -3451, 48101]\) | \(1771561/612\) | \(1570224562308\) | \([2]\) | \(411264\) | \(1.0419\) | \(\Gamma_0(N)\)-optimal |
139638.bd2 | 139638x2 | \([1, 1, 1, 10239, 349281]\) | \(46268279/46818\) | \(-120122179016562\) | \([2]\) | \(822528\) | \(1.3885\) |
Rank
sage: E.rank()
The elliptic curves in class 139638.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 139638.bd do not have complex multiplication.Modular form 139638.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.