Properties

Label 336.a
Number of curves $6$
Conductor $336$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("336.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 336.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
336.a1 336e5 [0, -1, 0, -12544, 544960] [4] 256  
336.a2 336e4 [0, -1, 0, -784, 8704] [2, 4] 128  
336.a3 336e3 [0, -1, 0, -624, -5760] [2] 128  
336.a4 336e6 [0, -1, 0, -544, 13888] [4] 256  
336.a5 336e2 [0, -1, 0, -64, 64] [2, 2] 64  
336.a6 336e1 [0, -1, 0, 16, 0] [2] 32 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 336.a have rank \(1\).

Modular form 336.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{3} - 2q^{5} + q^{7} + q^{9} - 4q^{11} - 2q^{13} + 2q^{15} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.