Properties

Label 2.625.adq_fcy
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:  $1 - 94 x + 3456 x^{2} - 58750 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0718378155570$, $\pm0.139601802087$
Angle rank:  $2$ (numerical)
Number field:  4.0.8126784.1
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
$A(\F_{q^r})$ 335238 151838006388 59596773873664950 23283012271506939887568 9094947305751037807560718278 3552713697941285029025620128513300 1387778781239120804259028520524703737398 542101086251036793717738897805538910981485568 211758236813701316071394921615565104395922813606550 82718061255304423072225402427767423298134376160192534228

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 532 388702 244108384 152587549222 95367434660752 59604645096519694 37252902996904763812 23283064365742782751102 14551915228375526355820804 9094947017729464412655512302

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.8126784.1.
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.dq_fcy$2$(not in LMFDB)