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Properties

Label 112710.cl
Number of curves $4$
Conductor $112710$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cl1")
 
E.isogeny_class()
 

Elliptic curves in class 112710.cl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
112710.cl1 112710cv4 \([1, 0, 0, -22143186, 40035774636]\) \(49745123032831462081/97939634471640\) \(2364024684893989043160\) \([2]\) \(13271040\) \(2.9893\)  
112710.cl2 112710cv3 \([1, 0, 0, -18559586, -30617026404]\) \(29291056630578924481/175463302795560\) \(4235257578195722393640\) \([2]\) \(13271040\) \(2.9893\)  
112710.cl3 112710cv2 \([1, 0, 0, -1855386, 162132516]\) \(29263955267177281/16463793153600\) \(397395943246747598400\) \([2, 2]\) \(6635520\) \(2.6427\)  
112710.cl4 112710cv1 \([1, 0, 0, 456614, 20175716]\) \(436192097814719/259683840000\) \(-6268136606184960000\) \([2]\) \(3317760\) \(2.2962\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 112710.cl do not have complex multiplication.

Modular form 112710.2.a.cl

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - 4 q^{7} + q^{8} + q^{9} - q^{10} + 4 q^{11} + q^{12} + q^{13} - 4 q^{14} - q^{15} + q^{16} + q^{18} + 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.