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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)
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.