Properties

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

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

Elliptic curves in class 336.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
336.a1 336e5 \([0, -1, 0, -12544, 544960]\) \(53297461115137/147\) \(602112\) \([4]\) \(256\) \(0.76735\)  
336.a2 336e4 \([0, -1, 0, -784, 8704]\) \(13027640977/21609\) \(88510464\) \([2, 4]\) \(128\) \(0.42078\)  
336.a3 336e3 \([0, -1, 0, -624, -5760]\) \(6570725617/45927\) \(188116992\) \([2]\) \(128\) \(0.42078\)  
336.a4 336e6 \([0, -1, 0, -544, 13888]\) \(-4354703137/17294403\) \(-70837874688\) \([4]\) \(256\) \(0.76735\)  
336.a5 336e2 \([0, -1, 0, -64, 64]\) \(7189057/3969\) \(16257024\) \([2, 2]\) \(64\) \(0.074205\)  
336.a6 336e1 \([0, -1, 0, 16, 0]\) \(103823/63\) \(-258048\) \([2]\) \(32\) \(-0.27237\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 336.a do not have complex multiplication.

Modular form 336.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.