Properties

Label 75810bu
Number of curves $6$
Conductor $75810$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("75810.bv1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75810bu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75810.bv6 75810bu1 [1, 0, 1, 3602, 116768] [2] 221184 \(\Gamma_0(N)\)-optimal
75810.bv5 75810bu2 [1, 0, 1, -25278, 1202656] [2, 2] 442368  
75810.bv4 75810bu3 [1, 0, 1, -133578, -17771504] [2] 884736  
75810.bv2 75810bu4 [1, 0, 1, -379058, 89789168] [2, 2] 884736  
75810.bv3 75810bu5 [1, 0, 1, -353788, 102282656] [2] 1769472  
75810.bv1 75810bu6 [1, 0, 1, -6064808, 5748247568] [2] 1769472  

Rank

sage: E.rank()
 

The elliptic curves in class 75810bu have rank \(1\).

Modular form 75810.2.a.bv

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.