Properties

Label 117975.m
Number of curves $6$
Conductor $117975$
CM no
Rank $1$
Graph

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Show commands: 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)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9} + q^{12} + q^{13} - q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.