Show commands:
SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 72450.ei
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.ei1 | 72450dc2 | \([1, -1, 1, -13970, 632457]\) | \(89332607016927/1060723384\) | \(3579941421000\) | \([2]\) | \(147456\) | \(1.2205\) | |
72450.ei2 | 72450dc1 | \([1, -1, 1, -170, 25257]\) | \(-160103007/81288256\) | \(-274347864000\) | \([2]\) | \(73728\) | \(0.87390\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.ei have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.ei do not have complex multiplication.Modular form 72450.2.a.ei
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.