Show commands:
SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 258b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
258.a1 | 258b1 | \([1, 1, 0, -1916, 31440]\) | \(778510269523657/1540767744\) | \(1540767744\) | \([2]\) | \(196\) | \(0.65081\) | \(\Gamma_0(N)\)-optimal |
258.a2 | 258b2 | \([1, 1, 0, -1276, 53584]\) | \(-230042158153417/1131994839168\) | \(-1131994839168\) | \([2]\) | \(392\) | \(0.99738\) |
Rank
sage: E.rank()
The elliptic curves in class 258b have rank \(0\).
Complex multiplication
The elliptic curves in class 258b do not have complex multiplication.Modular form 258.2.a.b
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.