Show commands:
SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 88200ep
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
88200.fx2 | 88200ep1 | \([0, 0, 0, -51450, -2100875]\) | \(55296/25\) | \(6809671181250000\) | \([2]\) | \(516096\) | \(1.7333\) | \(\Gamma_0(N)\)-optimal |
88200.fx1 | 88200ep2 | \([0, 0, 0, -694575, -222692750]\) | \(8503056/5\) | \(21790947780000000\) | \([2]\) | \(1032192\) | \(2.0799\) |
Rank
sage: E.rank()
The elliptic curves in class 88200ep have rank \(1\).
Complex multiplication
The elliptic curves in class 88200ep do not have complex multiplication.Modular form 88200.2.a.ep
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 Cremona 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 Cremona labels.