Properties

Label 20691o
Number of curves $3$
Conductor $20691$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("o1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 20691o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
20691.i3 20691o1 \([0, 0, 1, 726, -333]\) \(32768/19\) \(-24537891411\) \([]\) \(10800\) \(0.68308\) \(\Gamma_0(N)\)-optimal
20691.i2 20691o2 \([0, 0, 1, -10164, -419598]\) \(-89915392/6859\) \(-8858178799371\) \([]\) \(32400\) \(1.2324\)  
20691.i1 20691o3 \([0, 0, 1, -837804, -295162893]\) \(-50357871050752/19\) \(-24537891411\) \([]\) \(97200\) \(1.7817\)  

Rank

sage: E.rank()
 

The elliptic curves in class 20691o have rank \(1\).

Complex multiplication

The elliptic curves in class 20691o do not have complex multiplication.

Modular form 20691.2.a.o

sage: E.q_eigenform(10)
 
\(q - 2q^{4} - 3q^{5} + q^{7} + 4q^{13} + 4q^{16} - 3q^{17} - q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.