Show commands:
SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 40425ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40425.bt1 | 40425ca1 | \([0, 1, 1, -1143, -15991]\) | \(-56197120/3267\) | \(-9608982075\) | \([]\) | \(27216\) | \(0.67174\) | \(\Gamma_0(N)\)-optimal |
40425.bt2 | 40425ca2 | \([0, 1, 1, 6207, -25546]\) | \(8990228480/5314683\) | \(-15631678506675\) | \([]\) | \(81648\) | \(1.2211\) |
Rank
sage: E.rank()
The elliptic curves in class 40425ca have rank \(1\).
Complex multiplication
The elliptic curves in class 40425ca do not have complex multiplication.Modular form 40425.2.a.ca
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.