Abelian variety isogeny class 2.25.ar_er over $\F_{5^{2}}$

• Label: 2.25.ar_er
• Base field: $\F_{5^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 17 x + 121 x^{2} - 425 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.0882611637788$, $\pm0.235677669977$
Angle rank:	$2$ (numerical)
Number field:	4.0.8525.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$305$	$362645$	$243880745$	$152850878405$	$95407952922000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$579$	$15609$	$391299$	$9769774$	$244151859$	$6103501329$	$152587630659$	$3814696757649$	$95367441364174$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=(3a+1)x^6+(3a+1)x^5+(a+1)x^3+(4a+4)x^2+(4a+2)x+2a+4$
• $y^2=(a+4)x^6+(2a+2)x^5+(a+4)x^4+ax^3+x^2+2x+4a+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5^{2}}$

The endomorphism algebra of this simple isogeny class is 4.0.8525.1.

Base change

This is a primitive isogeny class.

Twists

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

Twist	Extension degree	Common base change
2.25.r_er	$2$	2.625.abv_cdl