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Properties

Label 112710cp
Number of curves $4$
Conductor $112710$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cp1")
 
E.isogeny_class()
 

Elliptic curves in class 112710cp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
112710.ck4 112710cp1 \([1, 0, 0, 270209, 42557321]\) \(90391899763439/84690294000\) \(-2044217815055286000\) \([2]\) \(2433024\) \(2.2007\) \(\Gamma_0(N)\)-optimal
112710.ck3 112710cp2 \([1, 0, 0, -1400211, 383657085]\) \(12577973014374481/4642947562500\) \(112069467153225562500\) \([2, 2]\) \(4866048\) \(2.5472\)  
112710.ck2 112710cp3 \([1, 0, 0, -9703181, -11358403089]\) \(4185743240664514801/113629394531250\) \(2742737350926269531250\) \([2]\) \(9732096\) \(2.8938\)  
112710.ck1 112710cp4 \([1, 0, 0, -19823961, 33962783835]\) \(35694515311673154481/10400566692750\) \(251044396185354924750\) \([2]\) \(9732096\) \(2.8938\)  

Rank

sage: E.rank()
 

The elliptic curves in class 112710cp have rank \(1\).

Complex multiplication

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

Modular form 112710.2.a.cp

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.