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Properties

Label 87360.ds
Number of curves $6$
Conductor $87360$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ds1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 87360.ds have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 87360.ds do not have complex multiplication.

Modular form 87360.2.a.ds

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{7} + q^{9} + 6 q^{11} - q^{13} - q^{15} + 6 q^{17} - 2 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 87360.ds

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
87360.ds1 87360bj5 \([0, -1, 0, -54545505, -155033545503]\) \(68463752473882049153689/1817088000000000\) \(476338716672000000000\) \([2]\) \(8957952\) \(3.0730\)  
87360.ds2 87360bj6 \([0, -1, 0, -52415585, -167699327775]\) \(-60752633741424905775769/11197265625000000000\) \(-2935296000000000000000000\) \([2]\) \(17915904\) \(3.4196\)  
87360.ds3 87360bj3 \([0, -1, 0, -1168545, 139637025]\) \(673163386034885929/357608625192000\) \(93744955442331648000\) \([2]\) \(2985984\) \(2.5237\)  
87360.ds4 87360bj1 \([0, -1, 0, -917985, 338839137]\) \(326355561310674169/465699780\) \(122080403128320\) \([2]\) \(995328\) \(1.9744\) \(\Gamma_0(N)\)-optimal
87360.ds5 87360bj2 \([0, -1, 0, -909665, 345273825]\) \(-317562142497484249/12339342574650\) \(-3234684619889049600\) \([2]\) \(1990656\) \(2.3209\)  
87360.ds6 87360bj4 \([0, -1, 0, 4455775, 1087897377]\) \(37321015309599759191/23553520979625000\) \(-6174414203682816000000\) \([2]\) \(5971968\) \(2.8703\)