Properties

Label 129360.gk
Number of curves $2$
Conductor $129360$
CM no
Rank $1$
Graph

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

Elliptic curves in class 129360.gk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.gk1 129360hm2 \([0, 1, 0, -501648520, 4324447450868]\) \(9937296563535244838593567/1587762000000\) \(2230691291136000000\) \([2]\) \(20643840\) \(3.3656\)  
129360.gk2 129360hm1 \([0, 1, 0, -31356040, 67548038900]\) \(2426796094451411844127/969756530688000\) \(1362438103146430464000\) \([2]\) \(10321920\) \(3.0190\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 129360.gk have rank \(1\).

Complex multiplication

The elliptic curves in class 129360.gk do not have complex multiplication.

Modular form 129360.2.a.gk

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