Abelian variety isogeny class 2.25.aq_ek over F52\F_{5^{2}}

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

Invariants

Base field:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 8 x + 25 x^{2} )^{2}$
	$1 - 16 x + 114 x^{2} - 400 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.204832764699$, $\pm0.204832764699$
Angle rank:	$1$ (numerical)
Jacobians:	$5$

This isogeny class is not simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$324$	$374544$	$246929796$	$153413222400$	$95489208772164$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$598$	$15802$	$392734$	$9778090$	$244187638$	$6103580122$	$152587231294$	$3814690378570$	$95367393027478$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.25.ai^2 and its endomorphism algebra is $\mathrm{M}_{2}($$\Q(\sqrt{-1})$$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{2}}$.

Subfield	Primitive Model
$\F_{5}$	2.5.a_ai

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.25.a_ao	$2$	2.625.abc_cdq
2.25.q_ek	$2$	2.625.abc_cdq
2.25.i_bn	$3$	(not in LMFDB)