Properties

Label 117117.m
Number of curves $2$
Conductor $117117$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 117117.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
117117.m1 117117bm2 \([1, -1, 1, -259199726, 1071065665986]\) \(249120591156760861/80068829743287\) \(618986504156147911252648179\) \([2]\) \(43130880\) \(3.8444\)  
117117.m2 117117bm1 \([1, -1, 1, 45996529, 114214367310]\) \(1392134518764179/1534746617019\) \(-11864635043123163482604423\) \([2]\) \(21565440\) \(3.4978\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 117117.2.a.m

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