Properties

Label 129360gk
Number of curves $6$
Conductor $129360$
CM no
Rank $1$
Graph

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

Elliptic curves in class 129360gk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
129360.fj6 129360gk1 [0, 1, 0, 27424, -507760716] [2] 2949120 \(\Gamma_0(N)\)-optimal
129360.fj5 129360gk2 [0, 1, 0, -9384496, -10872167020] [2, 2] 5898240  
129360.fj4 129360gk3 [0, 1, 0, -19948896, 18078514740] [2] 11796480  
129360.fj2 129360gk4 [0, 1, 0, -149410816, -702994261516] [2, 2] 11796480  
129360.fj3 129360gk5 [0, 1, 0, -148669936, -710310303340] [2] 23592960  
129360.fj1 129360gk6 [0, 1, 0, -2390572816, -44989251846316] [2] 23592960  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 129360.2.a.gk

sage: E.q_eigenform(10)
 
\( q + q^{3} - q^{5} + q^{9} + q^{11} + 2q^{13} - q^{15} - 2q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.