Show commands:
SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 481650fd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
481650.fd4 | 481650fd1 | \([1, 1, 1, 39440287, -202701007969]\) | \(89962967236397039/287450726400000\) | \(-21679214894438400000000000\) | \([2]\) | \(124416000\) | \(3.5447\) | \(\Gamma_0(N)\)-optimal* |
481650.fd3 | 481650fd2 | \([1, 1, 1, -371567713, -2375289295969]\) | \(75224183150104868881/11219310000000000\) | \(846147913777968750000000000\) | \([2]\) | \(248832000\) | \(3.8913\) | \(\Gamma_0(N)\)-optimal* |
481650.fd2 | 481650fd3 | \([1, 1, 1, -13948689713, -634092115867969]\) | \(-3979640234041473454886161/1471455901872240\) | \(-110975571722813201283750000\) | \([2]\) | \(622080000\) | \(4.3494\) | |
481650.fd1 | 481650fd4 | \([1, 1, 1, -223179055213, -40581608879320969]\) | \(16300610738133468173382620881/2228489100\) | \(168070175691904687500\) | \([2]\) | \(1244160000\) | \(4.6960\) |
Rank
sage: E.rank()
The elliptic curves in class 481650fd have rank \(1\).
Complex multiplication
The elliptic curves in class 481650fd do not have complex multiplication.Modular form 481650.2.a.fd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.