Properties

Label 58800gk
Number of curves $2$
Conductor $58800$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("gk1")
 
E.isogeny_class()
 

Elliptic curves in class 58800gk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
58800.f1 58800gk1 \([0, -1, 0, -427408, 61477312]\) \(393349474783/153600000\) \(3371827200000000000\) \([2]\) \(1290240\) \(2.2539\) \(\Gamma_0(N)\)-optimal
58800.f2 58800gk2 \([0, -1, 0, 1364592, 441381312]\) \(12801408679457/11250000000\) \(-246960000000000000000\) \([2]\) \(2580480\) \(2.6005\)  

Rank

sage: E.rank()
 

The elliptic curves in class 58800gk have rank \(1\).

Complex multiplication

The elliptic curves in class 58800gk do not have complex multiplication.

Modular form 58800.2.a.gk

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{9} - 6 q^{11} + 6 q^{13} - 4 q^{19} + O(q^{20})\) 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 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.