Properties

Label 1.625.abx
Base Field $\F_{5^{4}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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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.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

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
$A(\F_{q^r})$ 577 389475 244114852 152587347075 95367421115377 59604644599372800 37252902982572547777 23283064365396690982275 14551915228368607603662052 9094947017729362333147159875

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

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-11}) \).
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
1.625.bx$2$(not in LMFDB)