Properties

Label 336336.id
Number of curves $6$
Conductor $336336$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("336336.id1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 336336.id

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
336336.id1 336336id3 [0, 1, 0, -5381392, 4803169748] [2] 6291456  
336336.id2 336336id6 [0, 1, 0, -2359072, -1351256908] [2] 12582912  
336336.id3 336336id4 [0, 1, 0, -371632, 58235540] [2, 2] 6291456  
336336.id4 336336id2 [0, 1, 0, -336352, 74958260] [2, 2] 3145728  
336336.id5 336336id1 [0, 1, 0, -18832, 1420628] [2] 1572864 \(\Gamma_0(N)\)-optimal
336336.id6 336336id5 [0, 1, 0, 1051328, 398038388] [2] 12582912  

Rank

sage: E.rank()
 

The elliptic curves in class 336336.id have rank \(1\).

Modular form 336336.2.a.id

sage: E.q_eigenform(10)
 
\( q + q^{3} + 2q^{5} + q^{9} + q^{11} - q^{13} + 2q^{15} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.