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Properties

Label 117117.u
Number of curves $2$
Conductor $117117$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 117117.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
117117.u1 117117bn1 \([1, -1, 1, -351383, -64714386]\) \(620650477/124509\) \(962539241463552753\) \([2]\) \(2695680\) \(2.1667\) \(\Gamma_0(N)\)-optimal
117117.u2 117117bn2 \([1, -1, 1, 736132, -386618826]\) \(5706550403/11647251\) \(-90041170860545071167\) \([2]\) \(5391360\) \(2.5133\)  

Rank

sage: E.rank()
 

The elliptic curves in class 117117.u have rank \(0\).

Complex multiplication

The elliptic curves in class 117117.u do not have complex multiplication.

Modular form 117117.2.a.u

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + 4 q^{5} + q^{7} + 3 q^{8} - 4 q^{10} - q^{11} - q^{14} - q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.