Properties

Label 117117.t
Number of curves $2$
Conductor $117117$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("t1")
 
E.isogeny_class()
 

Elliptic curves in class 117117.t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
117117.t1 117117cb1 \([1, -1, 1, -149, -412]\) \(226981/77\) \(123324201\) \([2]\) \(31104\) \(0.25473\) \(\Gamma_0(N)\)-optimal
117117.t2 117117cb2 \([1, -1, 1, 436, -3220]\) \(5735339/5929\) \(-9495963477\) \([2]\) \(62208\) \(0.60130\)  

Rank

sage: E.rank()
 

The elliptic curves in class 117117.t have rank \(1\).

Complex multiplication

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

Modular form 117117.2.a.t

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.