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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 119130.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 119130.bf1 | 119130bi8 | \([1, 1, 1, -7734165990, -260031825512925]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(399149289349977203943008748480\) | \([2]\) | \(278691840\) | \(4.5085\) | |
| 119130.bf2 | 119130bi5 | \([1, 1, 1, -7719626715, -261064557298995]\) | \(1081411559614045490773061881/522522049500\) | \(24582510160653109500\) | \([2]\) | \(92897280\) | \(3.9592\) | |
| 119130.bf3 | 119130bi6 | \([1, 1, 1, -815095590, 2223083672355]\) | \(1272998045160051207059881/691293848290254950400\) | \(32522528122695387856179302400\) | \([2, 2]\) | \(139345920\) | \(4.1619\) | |
| 119130.bf4 | 119130bi3 | \([1, 1, 1, -630263590, 6082227966755]\) | \(588530213343917460371881/861551575695360000\) | \(40532452905526398812160000\) | \([4]\) | \(69672960\) | \(3.8153\) | |
| 119130.bf5 | 119130bi2 | \([1, 1, 1, -482479215, -4079239291995]\) | \(264020672568758737421881/5803468580250000\) | \(273029292213680450250000\) | \([2, 2]\) | \(46448640\) | \(3.6126\) | |
| 119130.bf6 | 119130bi4 | \([1, 1, 1, -465331715, -4382592284995]\) | \(-236859095231405581781881/39282983014374049500\) | \(-1848102544219262822265109500\) | \([2]\) | \(92897280\) | \(3.9592\) | |
| 119130.bf7 | 119130bi1 | \([1, 1, 1, -31229215, -58962791995]\) | \(71595431380957421881/9522562500000000\) | \(447997342190062500000000\) | \([4]\) | \(23224320\) | \(3.2660\) | \(\Gamma_0(N)\)-optimal |
| 119130.bf8 | 119130bi7 | \([1, 1, 1, 3146662810, 17496454656035]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-2126251566340391950574222623680\) | \([2]\) | \(278691840\) | \(4.5085\) |
Rank
sage: E.rank()
The elliptic curves in class 119130.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 119130.bf do not have complex multiplication.Modular form 119130.2.a.bf
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
