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Properties

Label 65520eb
Number of curves $6$
Conductor $65520$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("eb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 65520eb have rank \(1\).

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 \(\mathbb{F}_p\)
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 65520eb do not have complex multiplication.

Modular form 65520.2.a.eb

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - q^{7} - 6 q^{11} + q^{13} - 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 65520eb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
65520.ca4 65520eb1 \([0, 0, 0, -2065467, 1142549354]\) \(326355561310674169/465699780\) \(1390572091883520\) \([2]\) \(995328\) \(2.1771\) \(\Gamma_0(N)\)-optimal
65520.ca5 65520eb2 \([0, 0, 0, -2046747, 1164275786]\) \(-317562142497484249/12339342574650\) \(-36845079498423705600\) \([2]\) \(1990656\) \(2.5237\)  
65520.ca3 65520eb3 \([0, 0, 0, -2629227, 469960346]\) \(673163386034885929/357608625192000\) \(1067813633085308928000\) \([2]\) \(2985984\) \(2.7264\)  
65520.ca6 65520eb4 \([0, 0, 0, 10025493, 3676666394]\) \(37321015309599759191/23553520979625000\) \(-70330436788824576000000\) \([2]\) \(5971968\) \(3.0730\)  
65520.ca1 65520eb5 \([0, 0, 0, -122727387, -523299579766]\) \(68463752473882049153689/1817088000000000\) \(5425795694592000000000\) \([2]\) \(8957952\) \(3.2757\)  
65520.ca2 65520eb6 \([0, 0, 0, -117935067, -566044198774]\) \(-60752633741424905775769/11197265625000000000\) \(-33434856000000000000000000\) \([2]\) \(17915904\) \(3.6223\)