Properties

Label 8112k
Number of curves $6$
Conductor $8112$
CM no
Rank $0$
Graph

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

Elliptic curves in class 8112k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8112.be5 8112k1 \([0, 1, 0, 113, -532]\) \(2048/3\) \(-231686832\) \([2]\) \(2304\) \(0.29055\) \(\Gamma_0(N)\)-optimal
8112.be4 8112k2 \([0, 1, 0, -732, -5940]\) \(35152/9\) \(11120967936\) \([2, 2]\) \(4608\) \(0.63712\)  
8112.be2 8112k3 \([0, 1, 0, -10872, -439932]\) \(28756228/3\) \(14827957248\) \([2]\) \(9216\) \(0.98370\)  
8112.be3 8112k4 \([0, 1, 0, -4112, 95460]\) \(1556068/81\) \(400354845696\) \([2, 2]\) \(9216\) \(0.98370\)  
8112.be1 8112k5 \([0, 1, 0, -64952, 6349812]\) \(3065617154/9\) \(88967743488\) \([2]\) \(18432\) \(1.3303\)  
8112.be6 8112k6 \([0, 1, 0, 2648, 384788]\) \(207646/6561\) \(-64857485002752\) \([2]\) \(18432\) \(1.3303\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8112k have rank \(0\).

Complex multiplication

The elliptic curves in class 8112k do not have complex multiplication.

Modular form 8112.2.a.k

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