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Properties

Label 32175p
Number of curves $6$
Conductor $32175$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("p1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 32175p have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(5\)\(1\)
\(11\)\(1 + T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 32175p do not have complex multiplication.

Modular form 32175.2.a.p

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + 3 q^{8} + q^{11} - q^{13} - q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 32175p

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32175.j5 32175p1 \([1, -1, 1, -5405, -218028]\) \(-1532808577/938223\) \(-10686946359375\) \([2]\) \(65536\) \(1.2020\) \(\Gamma_0(N)\)-optimal
32175.j4 32175p2 \([1, -1, 1, -96530, -11517528]\) \(8732907467857/1656369\) \(18867078140625\) \([2, 2]\) \(131072\) \(1.5485\)  
32175.j3 32175p3 \([1, -1, 1, -106655, -8945778]\) \(11779205551777/3763454409\) \(42868097877515625\) \([2, 2]\) \(262144\) \(1.8951\)  
32175.j1 32175p4 \([1, -1, 1, -1544405, -738350778]\) \(35765103905346817/1287\) \(14659734375\) \([2]\) \(262144\) \(1.8951\)  
32175.j6 32175p5 \([1, -1, 1, 301720, -61217778]\) \(266679605718863/296110251723\) \(-3372880836032296875\) \([2]\) \(524288\) \(2.2417\)  
32175.j2 32175p6 \([1, -1, 1, -677030, 207796722]\) \(3013001140430737/108679952667\) \(1237932585847546875\) \([2]\) \(524288\) \(2.2417\)