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Properties

Label 9408bj
Number of curves $6$
Conductor $9408$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bj1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 9408bj have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 9408bj do not have complex multiplication.

Modular form 9408.2.a.bj

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} + q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{15} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 9408bj

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9408.bv6 9408bj1 \([0, 1, 0, 3071, 15455]\) \(103823/63\) \(-1942981705728\) \([2]\) \(12288\) \(1.0472\) \(\Gamma_0(N)\)-optimal
9408.bv5 9408bj2 \([0, 1, 0, -12609, 112671]\) \(7189057/3969\) \(122407847460864\) \([2, 2]\) \(24576\) \(1.3937\)  
9408.bv3 9408bj3 \([0, 1, 0, -122369, -16417185]\) \(6570725617/45927\) \(1416433663475712\) \([2]\) \(49152\) \(1.7403\)  
9408.bv2 9408bj4 \([0, 1, 0, -153729, 23115231]\) \(13027640977/21609\) \(666442725064704\) \([2, 2]\) \(49152\) \(1.7403\)  
9408.bv1 9408bj5 \([0, 1, 0, -2458689, 1483076895]\) \(53297461115137/147\) \(4533623980032\) \([2]\) \(98304\) \(2.0869\)  
9408.bv4 9408bj6 \([0, 1, 0, -106689, 37575327]\) \(-4354703137/17294403\) \(-533376327626784768\) \([2]\) \(98304\) \(2.0869\)