Properties

Label 9075.g
Number of curves 8
Conductor 9075
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("9075.g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 9075.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
9075.g1 9075l7 [1, 0, 0, -6534063, 6428154492] [2] 122880  
9075.g2 9075l5 [1, 0, 0, -408438, 100383867] [2, 2] 61440  
9075.g3 9075l8 [1, 0, 0, -332813, 138725742] [2] 122880  
9075.g4 9075l3 [1, 0, 0, -242063, -45859758] [2] 30720  
9075.g5 9075l4 [1, 0, 0, -30313, 936992] [2, 2] 30720  
9075.g6 9075l2 [1, 0, 0, -15188, -711633] [2, 2] 15360  
9075.g7 9075l1 [1, 0, 0, -63, -31008] [2] 7680 \(\Gamma_0(N)\)-optimal
9075.g8 9075l6 [1, 0, 0, 105812, 7062617] [2] 61440  

Rank

sage: E.rank()
 

The elliptic curves in class 9075.g have rank \(1\).

Modular form 9075.2.a.g

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.