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SageMath
E = EllipticCurve("gs1")
E.isogeny_class()
Elliptic curves in class 119952gs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
119952.bx5 | 119952gs1 | \([0, 0, 0, -495641811, -4241860146286]\) | \(38331145780597164097/55468445663232\) | \(19485955778292453224742912\) | \([2]\) | \(35389440\) | \(3.7548\) | \(\Gamma_0(N)\)-optimal |
119952.bx4 | 119952gs2 | \([0, 0, 0, -640148691, -1566719882350]\) | \(82582985847542515777/44772582831427584\) | \(15728520219044826202446495744\) | \([2, 2]\) | \(70778880\) | \(4.1014\) | |
119952.bx6 | 119952gs3 | \([0, 0, 0, 2469007149, -12322533595246]\) | \(4738217997934888496063/2928751705237796928\) | \(-1028864709142041498010731675648\) | \([2]\) | \(141557760\) | \(4.4480\) | |
119952.bx2 | 119952gs4 | \([0, 0, 0, -6061414611, 180398070722450]\) | \(70108386184777836280897/552468975892674624\) | \(194081263760003951884752912384\) | \([2, 2]\) | \(141557760\) | \(4.4480\) | |
119952.bx3 | 119952gs5 | \([0, 0, 0, -2064613971, 414744081367826]\) | \(-2770540998624539614657/209924951154647363208\) | \(-73746222127712565105575019184128\) | \([2]\) | \(283115520\) | \(4.7945\) | |
119952.bx1 | 119952gs6 | \([0, 0, 0, -96798469971, 11591798658784274]\) | \(285531136548675601769470657/17941034271597192\) | \(6302650024767289886032822272\) | \([2]\) | \(283115520\) | \(4.7945\) |
Rank
sage: E.rank()
The elliptic curves in class 119952gs have rank \(1\).
Complex multiplication
The elliptic curves in class 119952gs do not have complex multiplication.Modular form 119952.2.a.gs
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.