Properties

Label 117.a
Number of curves $4$
Conductor $117$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 117.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
117.a1 117a4 \([1, -1, 1, -626, 6180]\) \(37159393753/1053\) \(767637\) \([2]\) \(32\) \(0.23094\)  
117.a2 117a3 \([1, -1, 1, -176, -768]\) \(822656953/85683\) \(62462907\) \([2]\) \(32\) \(0.23094\)  
117.a3 117a2 \([1, -1, 1, -41, 96]\) \(10218313/1521\) \(1108809\) \([2, 2]\) \(16\) \(-0.11563\)  
117.a4 117a1 \([1, -1, 1, 4, 6]\) \(12167/39\) \(-28431\) \([4]\) \(8\) \(-0.46221\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 117.a have rank \(1\).

Complex multiplication

The elliptic curves in class 117.a do not have complex multiplication.

Modular form 117.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7} + 3 q^{8} + 2 q^{10} - 4 q^{11} + q^{13} + 4 q^{14} - q^{16} - 2 q^{17} + O(q^{20})\) 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.