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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 471510.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
471510.r1 | 471510r3 | \([1, -1, 0, -220618800, 1260371020000]\) | \(337492110729985137289/298819462500000\) | \(1051469119337248462500000\) | \([2]\) | \(82575360\) | \(3.5345\) | \(\Gamma_0(N)\)-optimal* |
471510.r2 | 471510r2 | \([1, -1, 0, -16926480, 10066821376]\) | \(152417923851548809/75773543040000\) | \(266627681820864973440000\) | \([2, 2]\) | \(41287680\) | \(3.1880\) | \(\Gamma_0(N)\)-optimal* |
471510.r3 | 471510r1 | \([1, -1, 0, -9138960, -10521824000]\) | \(23989788887201929/285238886400\) | \(1003682551914587750400\) | \([2]\) | \(20643840\) | \(2.8414\) | \(\Gamma_0(N)\)-optimal* |
471510.r4 | 471510r4 | \([1, -1, 0, 62165520, 77342476576]\) | \(7550657627997219191/5104288007114400\) | \(-17960681579380918613258400\) | \([2]\) | \(82575360\) | \(3.5345\) |
Rank
sage: E.rank()
The elliptic curves in class 471510.r have rank \(0\).
Complex multiplication
The elliptic curves in class 471510.r do not have complex multiplication.Modular form 471510.2.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 LMFDB 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 LMFDB labels.