Show commands:
SageMath
E = EllipticCurve("bs1")
E.isogeny_class()
Elliptic curves in class 127050.bs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 127050.bs1 | 127050bf8 | \([1, 1, 0, -1062478375, 13329499403125]\) | \(4791901410190533590281/41160000\) | \(1139335168125000000\) | \([2]\) | \(39813120\) | \(3.5059\) | |
| 127050.bs2 | 127050bf6 | \([1, 1, 0, -66406375, 208242947125]\) | \(1169975873419524361/108425318400\) | \(3001282273281600000000\) | \([2, 2]\) | \(19906560\) | \(3.1594\) | |
| 127050.bs3 | 127050bf7 | \([1, 1, 0, -61566375, 239891707125]\) | \(-932348627918877961/358766164249920\) | \(-9930877261011758205000000\) | \([2]\) | \(39813120\) | \(3.5059\) | |
| 127050.bs4 | 127050bf5 | \([1, 1, 0, -13181500, 18090887500]\) | \(9150443179640281/184570312500\) | \(5109024490356445312500\) | \([2]\) | \(13271040\) | \(2.9566\) | |
| 127050.bs5 | 127050bf3 | \([1, 1, 0, -4454375, 2748163125]\) | \(353108405631241/86318776320\) | \(2389359026503680000000\) | \([2]\) | \(9953280\) | \(2.8128\) | |
| 127050.bs6 | 127050bf2 | \([1, 1, 0, -1747000, -467306000]\) | \(21302308926361/8930250000\) | \(247195040941406250000\) | \([2, 2]\) | \(6635520\) | \(2.6101\) | |
| 127050.bs7 | 127050bf1 | \([1, 1, 0, -1505000, -711000000]\) | \(13619385906841/6048000\) | \(167412514500000000\) | \([2]\) | \(3317760\) | \(2.2635\) | \(\Gamma_0(N)\)-optimal |
| 127050.bs8 | 127050bf4 | \([1, 1, 0, 5815500, -3424243500]\) | \(785793873833639/637994920500\) | \(-17660108114935945312500\) | \([2]\) | \(13271040\) | \(2.9566\) |
Rank
sage: E.rank()
The elliptic curves in class 127050.bs have rank \(1\).
Complex multiplication
The elliptic curves in class 127050.bs do not have complex multiplication.Modular form 127050.2.a.bs
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
