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Properties

Label 82368.y
Number of curves $6$
Conductor $82368$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("y1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 82368.y have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(11\)\(1 + T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 82368.y do not have complex multiplication.

Modular form 82368.2.a.y

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 82368.y

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
82368.y1 82368w4 \([0, 0, 0, -3953676, -3025866256]\) \(35765103905346817/1287\) \(245949530112\) \([2]\) \(1048576\) \(2.1301\)  
82368.y2 82368w6 \([0, 0, 0, -1733196, 850442096]\) \(3013001140430737/108679952667\) \(20769062386202836992\) \([2]\) \(2097152\) \(2.4767\)  
82368.y3 82368w3 \([0, 0, 0, -273036, -36751120]\) \(11779205551777/3763454409\) \(719207337600221184\) \([2, 2]\) \(1048576\) \(2.1301\)  
82368.y4 82368w2 \([0, 0, 0, -247116, -47274640]\) \(8732907467857/1656369\) \(316537045254144\) \([2, 2]\) \(524288\) \(1.7835\)  
82368.y5 82368w1 \([0, 0, 0, -13836, -898576]\) \(-1532808577/938223\) \(-179297207451648\) \([2]\) \(262144\) \(1.4370\) \(\Gamma_0(N)\)-optimal
82368.y6 82368w5 \([0, 0, 0, 772404, -250439056]\) \(266679605718863/296110251723\) \(-56587550328374427648\) \([2]\) \(2097152\) \(2.4767\)