Show commands:
SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 185130.fr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
185130.fr1 | 185130w4 | \([1, -1, 1, -7123172, -7315545729]\) | \(30949975477232209/478125000\) | \(617483122678125000\) | \([2]\) | \(7864320\) | \(2.5488\) | |
185130.fr2 | 185130w2 | \([1, -1, 1, -458492, -107027841]\) | \(8253429989329/936360000\) | \(1209278947452840000\) | \([2, 2]\) | \(3932160\) | \(2.2022\) | |
185130.fr3 | 185130w1 | \([1, -1, 1, -110012, 12291711]\) | \(114013572049/15667200\) | \(20233686963916800\) | \([2]\) | \(1966080\) | \(1.8556\) | \(\Gamma_0(N)\)-optimal |
185130.fr4 | 185130w3 | \([1, -1, 1, 630508, -539143041]\) | \(21464092074671/109596256200\) | \(-141540054404617657800\) | \([2]\) | \(7864320\) | \(2.5488\) |
Rank
sage: E.rank()
The elliptic curves in class 185130.fr have rank \(0\).
Complex multiplication
The elliptic curves in class 185130.fr do not have complex multiplication.Modular form 185130.2.a.fr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.