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Properties

Label 262080kc
Number of curves $6$
Conductor $262080$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("kc1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 262080kc have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1 + T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 262080kc do not have complex multiplication.

Modular form 262080.2.a.kc

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - q^{7} + 4 q^{11} - q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 262080kc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
262080.kc5 262080kc1 \([0, 0, 0, -213132, 596662544]\) \(-5602762882081/801531494400\) \(-153175053937567334400\) \([2]\) \(9437184\) \(2.5526\) \(\Gamma_0(N)\)-optimal
262080.kc4 262080kc2 \([0, 0, 0, -12009612, 15889619216]\) \(1002404925316922401/9348917760000\) \(1786606006315253760000\) \([2, 2]\) \(18874368\) \(2.8992\)  
262080.kc2 262080kc3 \([0, 0, 0, -191721612, 1021773625616]\) \(4078208988807294650401/359723582400\) \(68744247134021222400\) \([2]\) \(37748736\) \(3.2457\)  
262080.kc3 262080kc4 \([0, 0, 0, -21041292, -11245160176]\) \(5391051390768345121/2833965225000000\) \(541579188378009600000000\) \([2, 2]\) \(37748736\) \(3.2457\)  
262080.kc6 262080kc5 \([0, 0, 0, 79758708, -87732200176]\) \(293623352309352854879/187320324116835000\) \(-35797471404011740200960000\) \([2]\) \(75497472\) \(3.5923\)  
262080.kc1 262080kc6 \([0, 0, 0, -266348172, -1671384001264]\) \(10934663514379917006241/12996826171875000\) \(2483732160000000000000000\) \([2]\) \(75497472\) \(3.5923\)