Properties

Label 1.625.abx
Base field $\F_{5^{4}}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $1$
L-polynomial:  $1 - 49 x + 625 x^{2}$
Frobenius angles:  $\pm0.0637685608585$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-11}) \)
Galois group:  $C_2$

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $577$ $389475$ $244114852$ $152587347075$ $95367421115377$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $577$ $389475$ $244114852$ $152587347075$ $95367421115377$ $59604644599372800$ $37252902982572547777$ $23283064365396690982275$ $14551915228368607603662052$ $9094947017729362333147159875$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-11}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
1.625.bx$2$(not in LMFDB)