Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 123627.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123627.i1 | 123627f2 | \([1, 1, 1, -1463778, -675934176]\) | \(4956477625/52983\) | \(3707769885022267407\) | \([2]\) | \(2580480\) | \(2.3789\) | |
123627.i2 | 123627f1 | \([1, 1, 1, -21463, -26315500]\) | \(-15625/4263\) | \(-298326312587998527\) | \([2]\) | \(1290240\) | \(2.0323\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 123627.i have rank \(0\).
Complex multiplication
The elliptic curves in class 123627.i do not have complex multiplication.Modular form 123627.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.