Properties

Label 112710cr
Number of curves $6$
Conductor $112710$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("112710.cn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 112710cr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
112710.cn6 112710cr1 [1, 0, 0, 4329, 44265] [2] 294912 \(\Gamma_0(N)\)-optimal
112710.cn5 112710cr2 [1, 0, 0, -18791, 363321] [2, 2] 589824  
112710.cn3 112710cr3 [1, 0, 0, -163291, -25155379] [2, 2] 1179648  
112710.cn2 112710cr4 [1, 0, 0, -244211, 46394085] [2] 1179648  
112710.cn4 112710cr5 [1, 0, 0, -33241, -64092349] [2] 2359296  
112710.cn1 112710cr6 [1, 0, 0, -2605341, -1618837209] [2] 2359296  

Rank

sage: E.rank()
 

The elliptic curves in class 112710cr have rank \(0\).

Modular form 112710.2.a.cn

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} + q^{13} - q^{15} + q^{16} + q^{18} + 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.