Show commands:
SageMath
E = EllipticCurve("jf1")
E.isogeny_class()
Elliptic curves in class 331200jf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
331200.jf5 | 331200jf1 | \([0, 0, 0, -6048300, -6281282000]\) | \(-8194759433281/965779200\) | \(-2883801238732800000000\) | \([2]\) | \(14155776\) | \(2.8536\) | \(\Gamma_0(N)\)-optimal |
331200.jf4 | 331200jf2 | \([0, 0, 0, -99360300, -381208898000]\) | \(36330796409313601/428490000\) | \(1279464284160000000000\) | \([2, 2]\) | \(28311552\) | \(3.2002\) | |
331200.jf3 | 331200jf3 | \([0, 0, 0, -101952300, -360270722000]\) | \(39248884582600321/3935264062500\) | \(11750635526400000000000000\) | \([2, 2]\) | \(56623104\) | \(3.5468\) | |
331200.jf1 | 331200jf4 | \([0, 0, 0, -1589760300, -24397514498000]\) | \(148809678420065817601/20700\) | \(61809868800000000\) | \([2]\) | \(56623104\) | \(3.5468\) | |
331200.jf2 | 331200jf5 | \([0, 0, 0, -371952300, 2365109278000]\) | \(1905890658841300321/293666194803750\) | \(876882559024880640000000000\) | \([2]\) | \(113246208\) | \(3.8933\) | |
331200.jf6 | 331200jf6 | \([0, 0, 0, 126575700, -1745607458000]\) | \(75108181893694559/484313964843750\) | \(-1446153750000000000000000000\) | \([2]\) | \(113246208\) | \(3.8933\) |
Rank
sage: E.rank()
The elliptic curves in class 331200jf have rank \(0\).
Complex multiplication
The elliptic curves in class 331200jf do not have complex multiplication.Modular form 331200.2.a.jf
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.