Properties

Label 116688y
Number of curves $4$
Conductor $116688$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("y1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 116688y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
116688.p4 116688y1 \([0, 1, 0, 14325376, 2291499603636]\) \(79374649975090937760383/553856914190911653543936\) \(-2268597920525974132915961856\) \([2]\) \(56733696\) \(3.9280\) \(\Gamma_0(N)\)-optimal
116688.p3 116688y2 \([0, 1, 0, -2523228544, 47840592467636]\) \(433744050935826360922067531137/9612122270219882316693504\) \(39371252818820637969176592384\) \([2, 2]\) \(113467392\) \(4.2746\)  
116688.p2 116688y3 \([0, 1, 0, -5492869504, -86174552631628]\) \(4474676144192042711273397261697/1806328356954994499451382272\) \(7398720950087657469752861786112\) \([2]\) \(226934784\) \(4.6211\)  
116688.p1 116688y4 \([0, 1, 0, -40154450304, 3097038281725620]\) \(1748094148784980747354970849498497/887694600425282263291392\) \(3635997083341956150441541632\) \([4]\) \(226934784\) \(4.6211\)  

Rank

sage: E.rank()
 

The elliptic curves in class 116688y have rank \(0\).

Complex multiplication

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

Modular form 116688.2.a.y

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{5} + q^{9} - q^{11} + q^{13} - 2q^{15} - q^{17} - 8q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.