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

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

Invariants

Base field:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 8 x + 25 x^{2} )( 1 - 6 x + 25 x^{2} )$
	$1 - 14 x + 98 x^{2} - 350 x^{3} + 625 x^{4}$
Frobenius angles:	$\pm0.204832764699$, $\pm0.295167235301$
Angle rank:	$1$ (numerical)
Jacobians:	$10$
Isomorphism classes:	43

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$360$	$391680$	$249224040$	$153413222400$	$95432942413800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$626$	$15948$	$392734$	$9772332$	$244140626$	$6103394988$	$152587231294$	$3814695666252$	$95367431640626$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{8}}$.

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

The isogeny class factors as 1.25.ai $\times$ 1.25.ag and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.25.ai : \(\Q(\sqrt{-1}) \).
• 1.25.ag : \(\Q(\sqrt{-1}) \).

Endomorphism algebra over $\overline{\F}_{5^{2}}$

The base change of $A$ to $\F_{5^{8}}$ is 1.390625.boo^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{4}}$
  The base change of $A$ to $\F_{5^{4}}$ is 1.625.ao $\times$ 1.625.o. The endomorphism algebra for each factor is:

  • 1.625.ao : \(\Q(\sqrt{-1}) \).
  • 1.625.o : \(\Q(\sqrt{-1}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.25.ac_c	$2$	2.625.a_boo
2.25.c_c	$2$	2.625.a_boo
2.25.o_du	$2$	2.625.a_boo