Properties

Label 90090.dr
Number of curves $8$
Conductor $90090$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("dr1")
 
E.isogeny_class()
 

Elliptic curves in class 90090.dr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
90090.dr1 90090dv8 \([1, -1, 1, -951773162, 11301756420449]\) \(130796627670002750950880364889/4007004103295286093000\) \(2921105991302263561797000\) \([6]\) \(47775744\) \(3.7919\)  
90090.dr2 90090dv6 \([1, -1, 1, -61988162, 160936392449]\) \(36134533748915083453404889/5565686539253841000000\) \(4057385487116050089000000\) \([2, 6]\) \(23887872\) \(3.4454\)  
90090.dr3 90090dv5 \([1, -1, 1, -20825177, -11524334569]\) \(1370131553911340548947529/714126686285699857170\) \(520598354302275195876930\) \([2]\) \(15925248\) \(3.2426\)  
90090.dr4 90090dv3 \([1, -1, 1, -16988162, -24499607551]\) \(743764321292317933404889/74603529000000000000\) \(54385972641000000000000\) \([6]\) \(11943936\) \(3.0988\)  
90090.dr5 90090dv2 \([1, -1, 1, -16560527, -25909851949]\) \(688999042618248810121129/779639711718968100\) \(568357349843127744900\) \([2, 2]\) \(7962624\) \(2.8961\)  
90090.dr6 90090dv1 \([1, -1, 1, -16556027, -25924653349]\) \(688437529087783927489129/882972090000\) \(643686653610000\) \([2]\) \(3981312\) \(2.5495\) \(\Gamma_0(N)\)-optimal
90090.dr7 90090dv4 \([1, -1, 1, -12367877, -39348133729]\) \(-286999819333751016766729/751553009101890965970\) \(-547882143635278514192130\) \([2]\) \(15925248\) \(3.2426\)  
90090.dr8 90090dv7 \([1, -1, 1, 107796838, 887480364449]\) \(190026536708029086053555111/576736012771479654093000\) \(-420440553310408667833797000\) \([6]\) \(47775744\) \(3.7919\)  

Rank

sage: E.rank()
 

The elliptic curves in class 90090.dr have rank \(0\).

Complex multiplication

The elliptic curves in class 90090.dr do not have complex multiplication.

Modular form 90090.2.a.dr

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} - q^{11} + q^{13} + q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.