Properties

Label 129360.fk
Number of curves $6$
Conductor $129360$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("fk1")
 
E.isogeny_class()
 

Elliptic curves in class 129360.fk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.fk1 129360cb6 \([0, 1, 0, -82559136, -288751969836]\) \(258286045443018193442/8440380939375\) \(2033668868375612160000\) \([2]\) \(12582912\) \(3.1826\)  
129360.fk2 129360cb4 \([0, 1, 0, -23324016, 43348202820]\) \(11647843478225136004/128410942275\) \(15469997002456550400\) \([2]\) \(6291456\) \(2.8360\)  
129360.fk3 129360cb3 \([0, 1, 0, -5384136, -4099699836]\) \(143279368983686884/22699269140625\) \(2734639426688400000000\) \([2, 2]\) \(6291456\) \(2.8360\)  
129360.fk4 129360cb2 \([0, 1, 0, -1494516, 640969020]\) \(12257375872392016/1191317675625\) \(35880277304219040000\) \([2, 2]\) \(3145728\) \(2.4895\)  
129360.fk5 129360cb1 \([0, 1, 0, 112929, 48143304]\) \(84611246065664/580054565475\) \(-1091885433177092400\) \([2]\) \(1572864\) \(2.1429\) \(\Gamma_0(N)\)-optimal
129360.fk6 129360cb5 \([0, 1, 0, 9556944, -22788002700]\) \(400647648358480318/1163177490234375\) \(-280261977187500000000000\) \([2]\) \(12582912\) \(3.1826\)  

Rank

sage: E.rank()
 

The elliptic curves in class 129360.fk have rank \(0\).

Complex multiplication

The elliptic curves in class 129360.fk do not have complex multiplication.

Modular form 129360.2.a.fk

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