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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 206910er
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 206910.bd7 | 206910er1 | \([1, -1, 0, -94206690, 308884644756]\) | \(71595431380957421881/9522562500000000\) | \(12298084451550562500000000\) | \([2]\) | \(61931520\) | \(3.5420\) | \(\Gamma_0(N)\)-optimal |
| 206910.bd5 | 206910er2 | \([1, -1, 0, -1455456690, 21372050394756]\) | \(264020672568758737421881/5803468580250000\) | \(7494993780490781012250000\) | \([2, 2]\) | \(123863040\) | \(3.8886\) | |
| 206910.bd4 | 206910er3 | \([1, -1, 0, -1901266065, -31868086259619]\) | \(588530213343917460371881/861551575695360000\) | \(1112666263652036341923840000\) | \([2]\) | \(185794560\) | \(4.0914\) | |
| 206910.bd2 | 206910er4 | \([1, -1, 0, -23287184190, 1367808913193256]\) | \(1081411559614045490773061881/522522049500\) | \(674820490025482465500\) | \([2]\) | \(247726080\) | \(4.2352\) | |
| 206910.bd6 | 206910er5 | \([1, -1, 0, -1403729190, 22961460596256]\) | \(-236859095231405581781881/39282983014374049500\) | \(-50732714289835151514070465500\) | \([2]\) | \(247726080\) | \(4.2352\) | |
| 206910.bd3 | 206910er6 | \([1, -1, 0, -2458834065, -11648774848419]\) | \(1272998045160051207059881/691293848290254950400\) | \(892783862233609683285283737600\) | \([2, 2]\) | \(371589120\) | \(4.4379\) | |
| 206910.bd1 | 206910er7 | \([1, -1, 0, -23331043665, 1362398004213021]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(10957144623237226438007958659520\) | \([2]\) | \(743178240\) | \(4.7845\) | |
| 206910.bd8 | 206910er8 | \([1, -1, 0, 9492287535, -91666314409059]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-58368251026369232092221626704320\) | \([2]\) | \(743178240\) | \(4.7845\) |
Rank
sage: E.rank()
The elliptic curves in class 206910er have rank \(1\).
Complex multiplication
The elliptic curves in class 206910er do not have complex multiplication.Modular form 206910.2.a.er
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.
