Properties

Label 159936cp
Number of curves $2$
Conductor $159936$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cp1")
 
E.isogeny_class()
 

Elliptic curves in class 159936cp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
159936.g1 159936cp1 \([0, -1, 0, -587477, 177562701]\) \(-11632923639808/318495051\) \(-613918707795542016\) \([]\) \(2654208\) \(2.1945\) \(\Gamma_0(N)\)-optimal
159936.g2 159936cp2 \([0, -1, 0, 2611243, 714627789]\) \(1021544365555712/705905647251\) \(-1360676347796404617216\) \([]\) \(7962624\) \(2.7438\)  

Rank

sage: E.rank()
 

The elliptic curves in class 159936cp have rank \(1\).

Complex multiplication

The elliptic curves in class 159936cp do not have complex multiplication.

Modular form 159936.2.a.cp

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.