Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("75810.cn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 75810.cn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
75810.cn1 75810co8 [1, 1, 1, -126794940, 549489304797] [2] 6967296  
75810.cn2 75810co6 [1, 1, 1, -7924860, 8582892765] [2, 2] 3483648  
75810.cn3 75810co7 [1, 1, 1, -7347260, 9887806685] [2] 6967296  
75810.cn4 75810co5 [1, 1, 1, -1573065, 745388547] [2] 2322432  
75810.cn5 75810co3 [1, 1, 1, -531580, 113151197] [2] 1741824  
75810.cn6 75810co2 [1, 1, 1, -208485, -19322085] [2, 2] 1161216  
75810.cn7 75810co1 [1, 1, 1, -179605, -29360773] [2] 580608 \(\Gamma_0(N)\)-optimal
75810.cn8 75810co4 [1, 1, 1, 694015, -140979085] [2] 2322432  

Rank

sage: E.rank()
 

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

Modular form 75810.2.a.cn

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.