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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 200640.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
200640.e1 | 200640ge7 | \([0, -1, 0, -1371154081, 19410669138721]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(2224096755832044555620843520\) | \([2]\) | \(148635648\) | \(4.0760\) | |
200640.e2 | 200640ge4 | \([0, -1, 0, -1368576481, 19487758493281]\) | \(1081411559614045490773061881/522522049500\) | \(136976020144128000\) | \([2]\) | \(49545216\) | \(3.5267\) | |
200640.e3 | 200640ge6 | \([0, -1, 0, -144504481, -165922487519]\) | \(1272998045160051207059881/691293848290254950400\) | \(181218534566200593717657600\) | \([2, 2]\) | \(74317824\) | \(3.7294\) | |
200640.e4 | 200640ge3 | \([0, -1, 0, -111736481, -453999082719]\) | \(588530213343917460371881/861551575695360000\) | \(225850576259084451840000\) | \([2]\) | \(37158912\) | \(3.3828\) | |
200640.e5 | 200640ge2 | \([0, -1, 0, -85536481, 304514237281]\) | \(264020672568758737421881/5803468580250000\) | \(1521344467501056000000\) | \([2, 2]\) | \(24772608\) | \(3.1801\) | |
200640.e6 | 200640ge5 | \([0, -1, 0, -82496481, 327157981281]\) | \(-236859095231405581781881/39282983014374049500\) | \(-10297798299320070832128000\) | \([2]\) | \(49545216\) | \(3.5267\) | |
200640.e7 | 200640ge1 | \([0, -1, 0, -5536481, 4402237281]\) | \(71595431380957421881/9522562500000000\) | \(2496282624000000000000\) | \([2]\) | \(12386304\) | \(2.8335\) | \(\Gamma_0(N)\)-optimal |
200640.e8 | 200640ge8 | \([0, -1, 0, 557857119, -1306136308959]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-11847670375366882968805048320\) | \([2]\) | \(148635648\) | \(4.0760\) |
Rank
sage: E.rank()
The elliptic curves in class 200640.e have rank \(0\).
Complex multiplication
The elliptic curves in class 200640.e do not have complex multiplication.Modular form 200640.2.a.e
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.