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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 31350bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31350.bt7 | 31350bi1 | \([1, 1, 1, -2162688, 1073413281]\) | \(71595431380957421881/9522562500000000\) | \(148790039062500000000\) | \([2]\) | \(1548288\) | \(2.5985\) | \(\Gamma_0(N)\)-optimal |
| 31350.bt5 | 31350bi2 | \([1, 1, 1, -33412688, 74323413281]\) | \(264020672568758737421881/5803468580250000\) | \(90679196566406250000\) | \([2, 2]\) | \(3096576\) | \(2.9451\) | |
| 31350.bt4 | 31350bi3 | \([1, 1, 1, -43647063, -110866899219]\) | \(588530213343917460371881/861551575695360000\) | \(13461743370240000000000\) | \([2]\) | \(4644864\) | \(3.1478\) | |
| 31350.bt6 | 31350bi4 | \([1, 1, 1, -32225188, 79852413281]\) | \(-236859095231405581781881/39282983014374049500\) | \(-613796609599594523437500\) | \([2]\) | \(6193152\) | \(3.2917\) | |
| 31350.bt2 | 31350bi5 | \([1, 1, 1, -534600188, 4757419413281]\) | \(1081411559614045490773061881/522522049500\) | \(8164407023437500\) | \([4]\) | \(6193152\) | \(3.2917\) | |
| 31350.bt3 | 31350bi6 | \([1, 1, 1, -56447063, -40543699219]\) | \(1272998045160051207059881/691293848290254950400\) | \(10801466379535233600000000\) | \([2, 2]\) | \(9289728\) | \(3.4944\) | |
| 31350.bt8 | 31350bi7 | \([1, 1, 1, 217912937, -318744739219]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-706176184139661965895000000\) | \([2]\) | \(18579456\) | \(3.8410\) | |
| 31350.bt1 | 31350bi8 | \([1, 1, 1, -535607063, 4738598140781]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(132566497077467713095000000\) | \([4]\) | \(18579456\) | \(3.8410\) |
Rank
sage: E.rank()
The elliptic curves in class 31350bi have rank \(1\).
Complex multiplication
The elliptic curves in class 31350bi do not have complex multiplication.Modular form 31350.2.a.bi
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
