Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 88445.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
88445.r1 | 88445bp2 | \([0, 1, 1, -14275, -493594]\) | \(7575076864/1953125\) | \(82951736328125\) | \([]\) | \(217728\) | \(1.3800\) | |
88445.r2 | 88445bp1 | \([0, 1, 1, -4965, 132969]\) | \(318767104/125\) | \(5308911125\) | \([]\) | \(72576\) | \(0.83073\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 88445.r have rank \(2\).
Complex multiplication
The elliptic curves in class 88445.r do not have complex multiplication.Modular form 88445.2.a.r
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.