Properties

Label 2.625.ads_fgp
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:  $1 - 96 x + 3551 x^{2} - 60000 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0329685986909$, $\pm0.123763616233$
Angle rank:  $2$ (numerical)
Number field:  4.0.1382544.1
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
$A(\F_{q^r})$ 334081 151763978113 59594380004433604 23282953389443494065033 9094946078572247752125285601 3552713675232313439523674692790800 1387778780856318213353534037100883967361 542101086245051551821881384742752396737449993 211758236813612898211370000594869234549155590049604 82718061255303156311153815763672325272882022543262277953

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 530 388512 244098578 152587163332 95367421792850 59604644715526374 37252902986628985010 23283064365485718569092 14551915228369450326745778 9094947017729325130802729952

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.1382544.1.
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.ds_fgp$2$(not in LMFDB)