Properties

Label 129360.gc
Number of curves $4$
Conductor $129360$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("gc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 129360.gc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.gc1 129360hn4 \([0, 1, 0, -12249820, -16506315400]\) \(6749703004355978704/5671875\) \(170826348000000\) \([2]\) \(2488320\) \(2.4665\)  
129360.gc2 129360hn3 \([0, 1, 0, -765445, -258221650]\) \(-26348629355659264/24169921875\) \(-45497074218750000\) \([2]\) \(1244160\) \(2.1199\)  
129360.gc3 129360hn2 \([0, 1, 0, -154660, -21608392]\) \(13584145739344/1195803675\) \(36015387279379200\) \([2]\) \(829440\) \(1.9172\)  
129360.gc4 129360hn1 \([0, 1, 0, 10715, -1564942]\) \(72268906496/606436875\) \(-1141547070510000\) \([2]\) \(414720\) \(1.5706\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 129360.2.a.gc

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.