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Properties

Label 8400cd
Number of curves $6$
Conductor $8400$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 8400cd have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 8400cd do not have complex multiplication.

Modular form 8400.2.a.cd

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{7} + q^{9} - 4 q^{11} + 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 8400cd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8400.bn6 8400cd1 \([0, 1, 0, 392, 788]\) \(103823/63\) \(-4032000000\) \([2]\) \(4096\) \(0.53235\) \(\Gamma_0(N)\)-optimal
8400.bn5 8400cd2 \([0, 1, 0, -1608, 4788]\) \(7189057/3969\) \(254016000000\) \([2, 2]\) \(8192\) \(0.87892\)  
8400.bn3 8400cd3 \([0, 1, 0, -15608, -751212]\) \(6570725617/45927\) \(2939328000000\) \([2]\) \(16384\) \(1.2255\)  
8400.bn2 8400cd4 \([0, 1, 0, -19608, 1048788]\) \(13027640977/21609\) \(1382976000000\) \([2, 2]\) \(16384\) \(1.2255\)  
8400.bn1 8400cd5 \([0, 1, 0, -313608, 67492788]\) \(53297461115137/147\) \(9408000000\) \([2]\) \(32768\) \(1.5721\)  
8400.bn4 8400cd6 \([0, 1, 0, -13608, 1708788]\) \(-4354703137/17294403\) \(-1106841792000000\) \([2]\) \(32768\) \(1.5721\)