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SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 150150.dx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 150150.dx1 | 150150ct8 | \([1, 1, 1, -2643814338, -52322946390969]\) | \(130796627670002750950880364889/4007004103295286093000\) | \(62609439113988845203125000\) | \([2]\) | \(143327232\) | \(4.0473\) | |
| 150150.dx2 | 150150ct6 | \([1, 1, 1, -172189338, -745075890969]\) | \(36134533748915083453404889/5565686539253841000000\) | \(86963852175841265625000000\) | \([2, 2]\) | \(71663616\) | \(3.7008\) | |
| 150150.dx3 | 150150ct5 | \([1, 1, 1, -57847713, 53353400781]\) | \(1370131553911340548947529/714126686285699857170\) | \(11158229473214060268281250\) | \([2]\) | \(47775744\) | \(3.4980\) | |
| 150150.dx4 | 150150ct3 | \([1, 1, 1, -47189338, 113424109031]\) | \(743764321292317933404889/74603529000000000000\) | \(1165680140625000000000000\) | \([2]\) | \(35831808\) | \(3.3542\) | |
| 150150.dx5 | 150150ct2 | \([1, 1, 1, -46001463, 119953018281]\) | \(688999042618248810121129/779639711718968100\) | \(12181870495608876562500\) | \([2, 2]\) | \(23887872\) | \(3.1515\) | |
| 150150.dx6 | 150150ct1 | \([1, 1, 1, -45988963, 120021543281]\) | \(688437529087783927489129/882972090000\) | \(13796438906250000\) | \([2]\) | \(11943936\) | \(2.8049\) | \(\Gamma_0(N)\)-optimal |
| 150150.dx7 | 150150ct4 | \([1, 1, 1, -34355213, 182167285781]\) | \(-286999819333751016766729/751553009101890965970\) | \(-11743015767217046343281250\) | \([2]\) | \(47775744\) | \(3.4980\) | |
| 150150.dx8 | 150150ct7 | \([1, 1, 1, 299435662, -4108705390969]\) | \(190026536708029086053555111/576736012771479654093000\) | \(-9011500199554369595203125000\) | \([2]\) | \(143327232\) | \(4.0473\) |
Rank
sage: E.rank()
The elliptic curves in class 150150.dx have rank \(1\).
Complex multiplication
The elliptic curves in class 150150.dx do not have complex multiplication.Modular form 150150.2.a.dx
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
