Properties

Label 10710.n
Number of curves 8
Conductor 10710
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("10710.n1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 10710.n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10710.n1 10710l8 [1, -1, 0, -119698794, 503716997808] [6] 1769472  
10710.n2 10710l7 [1, -1, 0, -78613794, -265421985192] [6] 1769472  
10710.n3 10710l4 [1, -1, 0, -78382134, -267080313420] [2] 589824  
10710.n4 10710l6 [1, -1, 0, -9156294, 4087006308] [2, 6] 884736  
10710.n5 10710l5 [1, -1, 0, -5143734, -3731921100] [2] 589824  
10710.n6 10710l2 [1, -1, 0, -4898934, -4172120460] [2, 2] 294912  
10710.n7 10710l1 [1, -1, 0, -290934, -71922060] [2] 147456 \(\Gamma_0(N)\)-optimal
10710.n8 10710l3 [1, -1, 0, 2093706, 489256308] [6] 442368  

Rank

sage: E.rank()
 

The elliptic curves in class 10710.n have rank \(1\).

Modular form 10710.2.a.n

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} + 2q^{13} - q^{14} + q^{16} + q^{17} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.