Properties

Label 2.625.adt_fin
Base Field $\F_{5^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $2$
L-polynomial:  $1 - 97 x + 3601 x^{2} - 60625 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0393703564648$, $\pm0.103462082915$
Angle rank:  $2$ (numerical)
Number field:  4.0.242525.2
Galois group:  $D_{4}$

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 333505 151727766245 59593254905124145 23282927578878967699205 9094945601308328900003722000 3552713668129240209344524792228805 1387778780781257479202795525636673703345 542101086244940089425493905699926338327175045 211758236813634468933743894360555893208887431791905 82718061255303923373223018404102505412827726732197888000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 529 388419 244093969 152586994179 95367416788374 59604644596356579 37252902984614088769 23283064365480931295619 14551915228370932655500129 9094947017729409470169161374

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.242525.2.
All geometric endomorphisms are defined over $\F_{5^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.625.dt_fin$2$(not in LMFDB)