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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 357390.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
357390.e1 | 357390e7 | \([1, -1, 0, -69607493910, 7020789681355060]\) | \(1087533321226184807035053481/8484255812957933638080\) | \(290979831936133381674453377641920\) | \([2]\) | \(2229534720\) | \(5.0578\) | |
357390.e2 | 357390e4 | \([1, -1, 0, -69476640435, 7048673570432425]\) | \(1081411559614045490773061881/522522049500\) | \(17920649907116116825500\) | \([2]\) | \(743178240\) | \(4.5085\) | |
357390.e3 | 357390e6 | \([1, -1, 0, -7335860310, -60030595013900]\) | \(1272998045160051207059881/691293848290254950400\) | \(23708923001444937747154711449600\) | \([2, 2]\) | \(1114767360\) | \(4.7112\) | |
357390.e4 | 357390e3 | \([1, -1, 0, -5672372310, -164225827474700]\) | \(588530213343917460371881/861551575695360000\) | \(29548158168128744734064640000\) | \([2]\) | \(557383680\) | \(4.3646\) | |
357390.e5 | 357390e2 | \([1, -1, 0, -4342312935, 110135118570925]\) | \(264020672568758737421881/5803468580250000\) | \(199038354023773048232250000\) | \([2, 2]\) | \(371589120\) | \(4.1619\) | |
357390.e6 | 357390e5 | \([1, -1, 0, -4187985435, 118325803709425]\) | \(-236859095231405581781881/39282983014374049500\) | \(-1347266754735842597431264825500\) | \([2]\) | \(743178240\) | \(4.5085\) | |
357390.e7 | 357390e1 | \([1, -1, 0, -281062935, 1591714320925]\) | \(71595431380957421881/9522562500000000\) | \(326590062456555562500000000\) | \([2]\) | \(185794560\) | \(3.8153\) | \(\Gamma_0(N)\)-optimal |
357390.e8 | 357390e8 | \([1, -1, 0, 28319965290, -472375955747660]\) | \(73240740785321709623685719/45195275784938365817280\) | \(-1550037391862145731968608292662720\) | \([2]\) | \(2229534720\) | \(5.0578\) |
Rank
sage: E.rank()
The elliptic curves in class 357390.e have rank \(0\).
Complex multiplication
The elliptic curves in class 357390.e do not have complex multiplication.Modular form 357390.2.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.