Show commands:
SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 32340.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32340.bg1 | 32340bf2 | \([0, 1, 0, -204836, -33091836]\) | \(31558509702736/2620631475\) | \(78928556134982400\) | \([2]\) | \(331776\) | \(1.9844\) | |
32340.bg2 | 32340bf1 | \([0, 1, 0, 13459, -2355900]\) | \(143225913344/1361505915\) | \(-2562876950301360\) | \([2]\) | \(165888\) | \(1.6378\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 32340.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 32340.bg do not have complex multiplication.Modular form 32340.2.a.bg
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.