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Properties

Label 12168q
Number of curves $6$
Conductor $12168$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("q1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 12168q have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 12168q do not have complex multiplication.

Modular form 12168.2.a.q

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{5} + 4 q^{11} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 12168q

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12168.j5 12168q1 \([0, 0, 0, 1014, -15379]\) \(2048/3\) \(-168899700528\) \([2]\) \(9216\) \(0.83986\) \(\Gamma_0(N)\)-optimal
12168.j4 12168q2 \([0, 0, 0, -6591, -153790]\) \(35152/9\) \(8107185625344\) \([2, 2]\) \(18432\) \(1.1864\)  
12168.j2 12168q3 \([0, 0, 0, -97851, -11780314]\) \(28756228/3\) \(10809580833792\) \([2]\) \(36864\) \(1.5330\)  
12168.j3 12168q4 \([0, 0, 0, -37011, 2614430]\) \(1556068/81\) \(291858682512384\) \([2, 2]\) \(36864\) \(1.5330\)  
12168.j1 12168q5 \([0, 0, 0, -584571, 172029494]\) \(3065617154/9\) \(64857485002752\) \([2]\) \(73728\) \(1.8796\)  
12168.j6 12168q6 \([0, 0, 0, 23829, 10365446]\) \(207646/6561\) \(-47281106567006208\) \([2]\) \(73728\) \(1.8796\)