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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("e1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 336.2.a.e

Copy content 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

Copy content 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 336e

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