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SageMath
E = EllipticCurve("bu1")
E.isogeny_class()
Elliptic curves in class 455175bu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
455175.bu4 | 455175bu1 | \([1, -1, 1, -253923980, -1557349093978]\) | \(6585576176607121/187425\) | \(51531003767225390625\) | \([2]\) | \(56623104\) | \(3.2921\) | \(\Gamma_0(N)\)-optimal* |
455175.bu3 | 455175bu2 | \([1, -1, 1, -254249105, -1553160833728]\) | \(6610905152742241/35128130625\) | \(9658198381072218837890625\) | \([2, 2]\) | \(113246208\) | \(3.6387\) | \(\Gamma_0(N)\)-optimal* |
455175.bu2 | 455175bu3 | \([1, -1, 1, -397629230, 396235345772]\) | \(25288177725059761/14387797265625\) | \(3955809711068411627197265625\) | \([2, 2]\) | \(226492416\) | \(3.9853\) | \(\Gamma_0(N)\)-optimal* |
455175.bu5 | 455175bu4 | \([1, -1, 1, -116070980, -3234512258728]\) | \(-629004249876241/16074715228425\) | \(-4419614304351308686616015625\) | \([2]\) | \(226492416\) | \(3.9853\) | |
455175.bu1 | 455175bu5 | \([1, -1, 1, -4664894855, 122397359564522]\) | \(40832710302042509761/91556816413125\) | \(25172813933572937715439453125\) | \([2]\) | \(452984832\) | \(4.3318\) | \(\Gamma_0(N)\)-optimal* |
455175.bu6 | 455175bu6 | \([1, -1, 1, 1575554395, 3154746053522]\) | \(1573196002879828319/926055908203125\) | \(-254611660633728504180908203125\) | \([2]\) | \(452984832\) | \(4.3318\) |
Rank
sage: E.rank()
The elliptic curves in class 455175bu have rank \(1\).
Complex multiplication
The elliptic curves in class 455175bu do not have complex multiplication.Modular form 455175.2.a.bu
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.