Show commands:
SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 212160.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212160.bp1 | 212160dz1 | \([0, -1, 0, -390801, 91349361]\) | \(402876451435348816/13746755117745\) | \(225226835849134080\) | \([2]\) | \(2949120\) | \(2.1017\) | \(\Gamma_0(N)\)-optimal |
212160.bp2 | 212160dz2 | \([0, -1, 0, 134079, 317992545]\) | \(4067455675907516/669098843633025\) | \(-43850061816333926400\) | \([2]\) | \(5898240\) | \(2.4482\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.bp do not have complex multiplication.Modular form 212160.2.a.bp
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.