Properties

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

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

Elliptic curves in class 12168q

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 \([0, 0, 0, 23829, 10365446]\) \(207646/6561\) \(-47281106567006208\) \([2]\) \(73728\) \(1.8796\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 12168.2.a.q

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

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

The vertices are labelled with Cremona labels.