Properties

Label 75810x
Number of curves 8
Conductor 75810
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("75810.q1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75810x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75810.q7 75810x1 [1, 1, 0, 75803, -6021491] [2] 884736 \(\Gamma_0(N)\)-optimal
75810.q6 75810x2 [1, 1, 0, -386277, -54355059] [2, 2] 1769472  
75810.q5 75810x3 [1, 1, 0, -2725557, 1692151389] [2, 2] 3538944  
75810.q4 75810x4 [1, 1, 0, -5440277, -4884968259] [2] 3538944  
75810.q8 75810x5 [1, 1, 0, 458463, 5412997161] [2] 7077888  
75810.q2 75810x6 [1, 1, 0, -43338057, 109794503889] [2, 2] 7077888  
75810.q3 75810x7 [1, 1, 0, -43067307, 111234406539] [2] 14155776  
75810.q1 75810x8 [1, 1, 0, -693408807, 7027717411239] [2] 14155776  

Rank

sage: E.rank()
 

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

Modular form 75810.2.a.q

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.