Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 412090k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
412090.k2 | 412090k1 | \([1, 0, 1, -2926698, -2266587172]\) | \(-115501303/25600\) | \(-614483623169762483200\) | \([2]\) | \(27095040\) | \(2.7094\) | \(\Gamma_0(N)\)-optimal* |
412090.k1 | 412090k2 | \([1, 0, 1, -49080778, -132347246244]\) | \(544737993463/20000\) | \(480065330601376940000\) | \([2]\) | \(54190080\) | \(3.0560\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 412090k have rank \(1\).
Complex multiplication
The elliptic curves in class 412090k do not have complex multiplication.Modular form 412090.2.a.k
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.