Show commands:
SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 54080bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54080.s1 | 54080bp1 | \([0, 1, 0, -13745, -615185]\) | \(3631696/65\) | \(5140358512640\) | \([2]\) | \(86016\) | \(1.2347\) | \(\Gamma_0(N)\)-optimal |
54080.s2 | 54080bp2 | \([0, 1, 0, -225, -1758977]\) | \(-4/4225\) | \(-1336493213286400\) | \([2]\) | \(172032\) | \(1.5813\) |
Rank
sage: E.rank()
The elliptic curves in class 54080bp have rank \(0\).
Complex multiplication
The elliptic curves in class 54080bp do not have complex multiplication.Modular form 54080.2.a.bp
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.