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Properties

Label 117975p
Number of curves $8$
Conductor $117975$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([1, 1, 1, -332813, -74037094]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 117975p have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1\)
\(11\)\(1\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 117975p do not have complex multiplication.

Modular form 117975.2.a.p

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9} + q^{12} + q^{13} - q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 117975p

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
117975.n6 117975p1 \([1, 1, 1, -332813, -74037094]\) \(147281603041/5265\) \(145738572890625\) \([2]\) \(737280\) \(1.8068\) \(\Gamma_0(N)\)-optimal
117975.n5 117975p2 \([1, 1, 1, -347938, -66958594]\) \(168288035761/27720225\) \(767313586269140625\) \([2, 2]\) \(1474560\) \(2.1534\)  
117975.n7 117975p3 \([1, 1, 1, 635187, -375659844]\) \(1023887723039/2798036865\) \(-77451452915566640625\) \([2]\) \(2949120\) \(2.4999\)  
117975.n4 117975p4 \([1, 1, 1, -1573063, 695069156]\) \(15551989015681/1445900625\) \(40023455580087890625\) \([2, 2]\) \(2949120\) \(2.4999\)  
117975.n8 117975p5 \([1, 1, 1, 1830062, 3295056656]\) \(24487529386319/183539412225\) \(-5080488510323956640625\) \([2]\) \(5898240\) \(2.8465\)  
117975.n2 117975p6 \([1, 1, 1, -24578188, 46889360156]\) \(59319456301170001/594140625\) \(16446193121337890625\) \([2, 2]\) \(5898240\) \(2.8465\)  
117975.n3 117975p7 \([1, 1, 1, -23988313, 49247680406]\) \(-55150149867714721/5950927734375\) \(-164725492000579833984375\) \([2]\) \(11796480\) \(3.1931\)  
117975.n1 117975p8 \([1, 1, 1, -393250063, 3001425766406]\) \(242970740812818720001/24375\) \(674715615234375\) \([2]\) \(11796480\) \(3.1931\)