Show commands:
SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 14450bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14450.q2 | 14450bc1 | \([1, -1, 1, -3480, 117147]\) | \(-60698457/40960\) | \(-3144320000000\) | \([]\) | \(59904\) | \(1.0986\) | \(\Gamma_0(N)\)-optimal |
14450.q1 | 14450bc2 | \([1, -1, 1, -3152730, -2153969853]\) | \(-45145776875761017/2441406250\) | \(-187416076660156250\) | \([]\) | \(778752\) | \(2.3811\) |
Rank
sage: E.rank()
The elliptic curves in class 14450bc have rank \(2\).
Complex multiplication
The elliptic curves in class 14450bc do not have complex multiplication.Modular form 14450.2.a.bc
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.
\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.