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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 465690r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
465690.r3 | 465690r1 | \([1, 0, 1, -599516679, -5492088957494]\) | \(506530866772858616168689/16163434718208000000\) | \(760423026304082101248000000\) | \([2]\) | \(265420800\) | \(3.9324\) | \(\Gamma_0(N)\)-optimal* |
465690.r2 | 465690r2 | \([1, 0, 1, -1453902599, 13675888280522]\) | \(7224504146467604173590769/2463409872562500000000\) | \(115893287718800540062500000000\) | \([2, 2]\) | \(530841600\) | \(4.2789\) | \(\Gamma_0(N)\)-optimal* |
465690.r1 | 465690r3 | \([1, 0, 1, -20857652599, 1159203434780522]\) | \(21330370319108709464713590769/4884813221669250750000\) | \(229810341533878192143660750000\) | \([2]\) | \(1061683200\) | \(4.6255\) | \(\Gamma_0(N)\)-optimal* |
465690.r4 | 465690r4 | \([1, 0, 1, 4279672681, 94902302557226]\) | \(184261146868096453165569551/189333915710449218750000\) | \(-8907380867777824401855468750000\) | \([2]\) | \(1061683200\) | \(4.6255\) |
Rank
sage: E.rank()
The elliptic curves in class 465690r have rank \(0\).
Complex multiplication
The elliptic curves in class 465690r do not have complex multiplication.Modular form 465690.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.