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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 450450.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450450.e1 | 450450e7 | \([1, -1, 0, -23794329042, 1412695758227116]\) | \(130796627670002750950880364889/4007004103295286093000\) | \(45642281114097868153078125000\) | \([2]\) | \(1146617856\) | \(4.5967\) | \(\Gamma_0(N)\)-optimal* |
450450.e2 | 450450e6 | \([1, -1, 0, -1549704042, 20115499352116]\) | \(36134533748915083453404889/5565686539253841000000\) | \(63396648236188282640625000000\) | \([2, 2]\) | \(573308928\) | \(4.2501\) | \(\Gamma_0(N)\)-optimal* |
450450.e3 | 450450e4 | \([1, -1, 0, -520629417, -1441062450509]\) | \(1370131553911340548947529/714126686285699857170\) | \(8134349285973049935577031250\) | \([2]\) | \(382205952\) | \(4.0473\) | \(\Gamma_0(N)\)-optimal* |
450450.e4 | 450450e3 | \([1, -1, 0, -424704042, -3062875647884]\) | \(743764321292317933404889/74603529000000000000\) | \(849780822515625000000000000\) | \([2]\) | \(286654464\) | \(3.9035\) | \(\Gamma_0(N)\)-optimal* |
450450.e5 | 450450e2 | \([1, -1, 0, -414013167, -3239145506759]\) | \(688999042618248810121129/779639711718968100\) | \(8880583591298871014062500\) | \([2, 2]\) | \(191102976\) | \(3.7008\) | \(\Gamma_0(N)\)-optimal* |
450450.e6 | 450450e1 | \([1, -1, 0, -413900667, -3240995569259]\) | \(688437529087783927489129/882972090000\) | \(10057603962656250000\) | \([2]\) | \(95551488\) | \(3.3542\) | \(\Gamma_0(N)\)-optimal* |
450450.e7 | 450450e5 | \([1, -1, 0, -309196917, -4918825913009]\) | \(-286999819333751016766729/751553009101890965970\) | \(-8560658494301226784252031250\) | \([2]\) | \(382205952\) | \(4.0473\) | |
450450.e8 | 450450e8 | \([1, -1, 0, 2694920958, 110937740477116]\) | \(190026536708029086053555111/576736012771479654093000\) | \(-6569383645475135434903078125000\) | \([2]\) | \(1146617856\) | \(4.5967\) |
Rank
sage: E.rank()
The elliptic curves in class 450450.e have rank \(0\).
Complex multiplication
The elliptic curves in class 450450.e do not have complex multiplication.Modular form 450450.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.