Properties

Label 2.625.adp_fay
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:  $1 - 93 x + 3404 x^{2} - 58125 x^{3} + 390625 x^{4}$
Frobenius angles:  $\pm0.0504914747344$, $\pm0.162461973402$
Angle rank:  $2$ (numerical)
Number field:  4.0.10132056.2
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
$A(\F_{q^r})$ 335812 151870305376 59597561313658624 23283022426840033895808 9094947258961836567512982532 3552713690925025742425014646841344 1387778780992520692511845470737038473028 542101086244822113288969272127315005757838848 211758236813572410271651503459764619979749293821696 82718061255302132335600696260157150249487961486466612576

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 533 388785 244111610 152587615777 95367434170133 59604644978806398 37252902990285142469 23283064365475864259137 14551915228366668016714202 9094947017729212543500008625

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is 4.0.10132056.2.
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.dp_fay$2$(not in LMFDB)