Properties

Label 75810cy
Number of curves $4$
Conductor $75810$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cy1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75810cy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75810.cr3 75810cy1 [1, 0, 0, -1271, 10905] [2] 115200 \(\Gamma_0(N)\)-optimal
75810.cr2 75810cy2 [1, 0, 0, -8491, -293779] [2, 2] 230400  
75810.cr4 75810cy3 [1, 0, 0, 2339, -989065] [2] 460800  
75810.cr1 75810cy4 [1, 0, 0, -134841, -19069389] [2] 460800  

Rank

sage: E.rank()
 

The elliptic curves in class 75810cy have rank \(0\).

Complex multiplication

The elliptic curves in class 75810cy do not have complex multiplication.

Modular form 75810.2.a.cy

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} + q^{18} + O(q^{20}) \)

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 Cremona 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 Cremona labels.