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Properties

Label 69360.n
Number of curves $8$
Conductor $69360$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("n1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 69360.n have rank \(2\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 69360.n do not have complex multiplication.

Modular form 69360.2.a.n

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 69360.n

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
69360.n1 69360cc8 \([0, -1, 0, -9987936, 12152923200]\) \(1114544804970241/405\) \(40041330462720\) \([2]\) \(1310720\) \(2.4006\)  
69360.n2 69360cc6 \([0, -1, 0, -624336, 189987840]\) \(272223782641/164025\) \(16216738837401600\) \([2, 2]\) \(655360\) \(2.0541\)  
69360.n3 69360cc7 \([0, -1, 0, -508736, 262399680]\) \(-147281603041/215233605\) \(-21279604702438379520\) \([2]\) \(1310720\) \(2.4006\)  
69360.n4 69360cc4 \([0, -1, 0, -370016, -86508864]\) \(56667352321/15\) \(1483012239360\) \([2]\) \(327680\) \(1.7075\)  
69360.n5 69360cc3 \([0, -1, 0, -46336, 1791040]\) \(111284641/50625\) \(5005166307840000\) \([2, 2]\) \(327680\) \(1.7075\)  
69360.n6 69360cc2 \([0, -1, 0, -23216, -1334784]\) \(13997521/225\) \(22245183590400\) \([2, 2]\) \(163840\) \(1.3609\)  
69360.n7 69360cc1 \([0, -1, 0, -96, -58560]\) \(-1/15\) \(-1483012239360\) \([2]\) \(81920\) \(1.0143\) \(\Gamma_0(N)\)-optimal
69360.n8 69360cc5 \([0, -1, 0, 161744, 13277056]\) \(4733169839/3515625\) \(-347580993600000000\) \([2]\) \(655360\) \(2.0541\)