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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ff1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 87360.ff 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 \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 87360.2.a.ff

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + q^{7} + q^{9} + 4 q^{11} - q^{13} - q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 87360.ff

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
87360.ff1 87360ck6 \([0, 1, 0, -29594241, -61912975905]\) \(10934663514379917006241/12996826171875000\) \(3407040000000000000000\) \([2]\) \(9437184\) \(3.0430\)  
87360.ff2 87360ck4 \([0, 1, 0, -21302401, 37836366815]\) \(4078208988807294650401/359723582400\) \(94299378784665600\) \([2]\) \(4718592\) \(2.6964\)  
87360.ff3 87360ck3 \([0, 1, 0, -2337921, -417266721]\) \(5391051390768345121/2833965225000000\) \(742906979942400000000\) \([2, 2]\) \(4718592\) \(2.6964\)  
87360.ff4 87360ck2 \([0, 1, 0, -1334401, 588059615]\) \(1002404925316922401/9348917760000\) \(2450762697277440000\) \([2, 2]\) \(2359296\) \(2.3498\)  
87360.ff5 87360ck1 \([0, 1, 0, -23681, 22090719]\) \(-5602762882081/801531494400\) \(-210116672067993600\) \([2]\) \(1179648\) \(2.0033\) \(\Gamma_0(N)\)-optimal
87360.ff6 87360ck5 \([0, 1, 0, 8862079, -3246386721]\) \(293623352309352854879/187320324116835000\) \(-49104899045283594240000\) \([2]\) \(9437184\) \(3.0430\)