Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("117975.m1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 117975.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
117975.m1 117975q4 [1, 1, 1, -20763663, -36425643594] [2] 3932160  
117975.m2 117975q6 [1, 1, 1, -9102288, 10231472406] [2] 7864320  
117975.m3 117975q3 [1, 1, 1, -1433913, -442905594] [2, 2] 3932160  
117975.m4 117975q2 [1, 1, 1, -1297788, -569501844] [2, 2] 1966080  
117975.m5 117975q1 [1, 1, 1, -72663, -10844844] [2] 983040 \(\Gamma_0(N)\)-optimal
117975.m6 117975q5 [1, 1, 1, 4056462, -3012401094] [2] 7864320  

Rank

sage: E.rank()
 

The elliptic curves in class 117975.m have rank \(1\).

Modular form 117975.2.a.m

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{8} + q^{9} + q^{12} + q^{13} - q^{16} - 6q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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.