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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 117975.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117975.m1 | 117975q4 | \([1, 1, 1, -20763663, -36425643594]\) | \(35765103905346817/1287\) | \(35624984484375\) | \([2]\) | \(3932160\) | \(2.5448\) | |
117975.m2 | 117975q6 | \([1, 1, 1, -9102288, 10231472406]\) | \(3013001140430737/108679952667\) | \(3008330712917237296875\) | \([2]\) | \(7864320\) | \(2.8913\) | |
117975.m3 | 117975q3 | \([1, 1, 1, -1433913, -442905594]\) | \(11779205551777/3763454409\) | \(104174829004100765625\) | \([2, 2]\) | \(3932160\) | \(2.5448\) | |
117975.m4 | 117975q2 | \([1, 1, 1, -1297788, -569501844]\) | \(8732907467857/1656369\) | \(45849355031390625\) | \([2, 2]\) | \(1966080\) | \(2.1982\) | |
117975.m5 | 117975q1 | \([1, 1, 1, -72663, -10844844]\) | \(-1532808577/938223\) | \(-25970613689109375\) | \([2]\) | \(983040\) | \(1.8516\) | \(\Gamma_0(N)\)-optimal |
117975.m6 | 117975q5 | \([1, 1, 1, 4056462, -3012401094]\) | \(266679605718863/296110251723\) | \(-8196521463322650046875\) | \([2]\) | \(7864320\) | \(2.8913\) |
Rank
sage: E.rank()
The elliptic curves in class 117975.m have rank \(1\).
Complex multiplication
The elliptic curves in class 117975.m do not have complex multiplication.Modular form 117975.2.a.m
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.