Properties

Label 12696.k
Number of curves $6$
Conductor $12696$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12696.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.k1 12696j5 \([0, -1, 0, -203312, 35352972]\) \(3065617154/9\) \(2728597506048\) \([2]\) \(45056\) \(1.6155\)  
12696.k2 12696j3 \([0, -1, 0, -34032, -2404932]\) \(28756228/3\) \(454766251008\) \([2]\) \(22528\) \(1.2690\)  
12696.k3 12696j4 \([0, -1, 0, -12872, 540540]\) \(1556068/81\) \(12278688777216\) \([2, 2]\) \(22528\) \(1.2690\)  
12696.k4 12696j2 \([0, -1, 0, -2292, -30780]\) \(35152/9\) \(341074688256\) \([2, 2]\) \(11264\) \(0.92239\)  
12696.k5 12696j1 \([0, -1, 0, 353, -3272]\) \(2048/3\) \(-7105722672\) \([2]\) \(5632\) \(0.57582\) \(\Gamma_0(N)\)-optimal
12696.k6 12696j6 \([0, -1, 0, 8288, 2123308]\) \(207646/6561\) \(-1989147581908992\) \([2]\) \(45056\) \(1.6155\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12696.k have rank \(1\).

Complex multiplication

The elliptic curves in class 12696.k do not have complex multiplication.

Modular form 12696.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2q^{5} + q^{9} - 4q^{11} - 2q^{13} - 2q^{15} - 2q^{17} + 4q^{19} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.