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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("kr1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 262080kr 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.e
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 262080.2.a.kr

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} + q^{7} - 4 q^{11} - q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 262080kr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
262080.kr5 262080kr1 \([0, 0, 0, -26472, -2165816]\) \(-2748251600896/1124136195\) \(-839163173022720\) \([2]\) \(1048576\) \(1.5720\) \(\Gamma_0(N)\)-optimal
262080.kr4 262080kr2 \([0, 0, 0, -458652, -119545904]\) \(893359210685776/91298025\) \(1090457767526400\) \([2, 2]\) \(2097152\) \(1.9186\)  
262080.kr3 262080kr3 \([0, 0, 0, -493932, -100085456]\) \(278944461825124/70849130625\) \(3384869927362560000\) \([2, 2]\) \(4194304\) \(2.2651\)  
262080.kr1 262080kr4 \([0, 0, 0, -7338252, -7651331984]\) \(914732517663095044/9555\) \(456497233920\) \([2]\) \(4194304\) \(2.2651\)  
262080.kr2 262080kr5 \([0, 0, 0, -2761932, 1684376944]\) \(24385137179326562/1284775885575\) \(122762247613208985600\) \([2]\) \(8388608\) \(2.6117\)  
262080.kr6 262080kr6 \([0, 0, 0, 1209588, -639079184]\) \(2048324060764798/3031899609375\) \(-289702519142400000000\) \([2]\) \(8388608\) \(2.6117\)