Properties

Label 180336.bg
Number of curves $6$
Conductor $180336$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("180336.bg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 180336.bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
180336.bg1 180336bp5 [0, -1, 0, -93284672, 346765316928] [2] 18874368  
180336.bg2 180336bp3 [0, -1, 0, -6422832, 4251709440] [2, 2] 9437184  
180336.bg3 180336bp2 [0, -1, 0, -2515552, -1484177600] [2, 2] 4718592  
180336.bg4 180336bp1 [0, -1, 0, -2492432, -1513715712] [2] 2359296 \(\Gamma_0(N)\)-optimal
180336.bg5 180336bp4 [0, -1, 0, 1021808, -5329995392] [4] 9437184  
180336.bg6 180336bp6 [0, -1, 0, 17922528, 28772356032] [2] 18874368  

Rank

sage: E.rank()
 

The elliptic curves in class 180336.bg have rank \(1\).

Modular form 180336.2.a.bg

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.