Show commands:
SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 12870.be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.be1 | 12870bf2 | \([1, -1, 1, -22508, 1305327]\) | \(46703838741180867/148720000\) | \(4015440000\) | \([2]\) | \(21504\) | \(1.0674\) | |
| 12870.be2 | 12870bf1 | \([1, -1, 1, -1388, 21231]\) | \(-10945484159427/644300800\) | \(-17396121600\) | \([2]\) | \(10752\) | \(0.72082\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.be have rank \(1\).
Complex multiplication
The elliptic curves in class 12870.be do not have complex multiplication.Modular form 12870.2.a.be
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.
