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Properties

Label 49686.cp
Number of curves $3$
Conductor $49686$
CM no
Rank $1$
Graph

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

Elliptic curves in class 49686.cp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
49686.cp1 49686cj3 \([1, 1, 1, -215774049, 1219893596319]\) \(-1956469094246217097/36641439744\) \(-20807546981130650517504\) \([]\) \(15676416\) \(3.4069\)  
49686.cp2 49686cj2 \([1, 1, 1, -1006314, 3720430839]\) \(-198461344537/10417365504\) \(-5915701556994194993664\) \([]\) \(5225472\) \(2.8576\)  
49686.cp3 49686cj1 \([1, 1, 1, 111621, -136444911]\) \(270840023/14329224\) \(-8137125715207946184\) \([]\) \(1741824\) \(2.3083\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 49686.cp have rank \(1\).

Complex multiplication

The elliptic curves in class 49686.cp do not have complex multiplication.

Modular form 49686.2.a.cp

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + 3 q^{5} - q^{6} + q^{8} + q^{9} + 3 q^{10} - 3 q^{11} - q^{12} - 3 q^{15} + q^{16} + 3 q^{17} + q^{18} - 7 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.