Properties

Label 116688u
Number of curves $2$
Conductor $116688$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 116688u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
116688.y1 116688u1 \([0, 1, 0, -1349408, 602853876]\) \(66342819962001390625/4812668669952\) \(19712690872123392\) \([2]\) \(1419264\) \(2.1778\) \(\Gamma_0(N)\)-optimal
116688.y2 116688u2 \([0, 1, 0, -1262368, 684079604]\) \(-54315282059491182625/17983956399469632\) \(-73662285412227612672\) \([2]\) \(2838528\) \(2.5244\)  

Rank

sage: E.rank()
 

The elliptic curves in class 116688u have rank \(1\).

Complex multiplication

The elliptic curves in class 116688u do not have complex multiplication.

Modular form 116688.2.a.u

sage: E.q_eigenform(10)
 
\(q + q^{3} + 4 q^{7} + q^{9} - q^{11} - q^{13} - q^{17} - 2 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.