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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 18810.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18810.p1 | 18810z7 | \([1, -1, 1, -192818543, -1023537198409]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(6185022487646333622160320\) | \([6]\) | \(6193152\) | \(3.5856\) | |
| 18810.p2 | 18810z4 | \([1, -1, 1, -192456068, -1027602593269]\) | \(1081411559614045490773061881/522522049500\) | \(380918574085500\) | \([2]\) | \(2064384\) | \(3.0362\) | |
| 18810.p3 | 18810z6 | \([1, -1, 1, -20320943, 8757439031]\) | \(1272998045160051207059881/691293848290254950400\) | \(503953215403595858841600\) | \([2, 6]\) | \(3096576\) | \(3.2390\) | |
| 18810.p4 | 18810z3 | \([1, -1, 1, -15712943, 23947250231]\) | \(588530213343917460371881/861551575695360000\) | \(628071098681917440000\) | \([6]\) | \(1548288\) | \(2.8924\) | |
| 18810.p5 | 18810z2 | \([1, -1, 1, -12028568, -16053857269]\) | \(264020672568758737421881/5803468580250000\) | \(4230728595002250000\) | \([2, 2]\) | \(1032192\) | \(2.6897\) | |
| 18810.p6 | 18810z5 | \([1, -1, 1, -11601068, -17248121269]\) | \(-236859095231405581781881/39282983014374049500\) | \(-28637294617478682085500\) | \([2]\) | \(2064384\) | \(3.0362\) | |
| 18810.p7 | 18810z1 | \([1, -1, 1, -778568, -231857269]\) | \(71595431380957421881/9522562500000000\) | \(6941948062500000000\) | \([2]\) | \(516096\) | \(2.3431\) | \(\Gamma_0(N)\)-optimal |
| 18810.p8 | 18810z8 | \([1, -1, 1, 78448657, 68848863671]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-32947356047220068680797120\) | \([6]\) | \(6193152\) | \(3.5856\) |
Rank
sage: E.rank()
The elliptic curves in class 18810.p have rank \(1\).
Complex multiplication
The elliptic curves in class 18810.p do not have complex multiplication.Modular form 18810.2.a.p
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.
