Show commands:
SageMath
E = EllipticCurve("em1")
E.isogeny_class()
Elliptic curves in class 286110em
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
286110.em1 | 286110em1 | \([1, -1, 1, -22117658, -40030515723]\) | \(68001744211490809/1022422500\) | \(17990840564217922500\) | \([2]\) | \(18579456\) | \(2.8313\) | \(\Gamma_0(N)\)-optimal |
286110.em2 | 286110em2 | \([1, -1, 1, -21467408, -42495223323]\) | \(-62178675647294809/8362782148050\) | \(-147153921494152790938050\) | \([2]\) | \(37158912\) | \(3.1779\) |
Rank
sage: E.rank()
The elliptic curves in class 286110em have rank \(1\).
Complex multiplication
The elliptic curves in class 286110em do not have complex multiplication.Modular form 286110.2.a.em
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.