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

• Label: 2.25.aq_eg
• Base field: $\F_{5^{2}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 - 5 x )^{2}( 1 - 6 x + 25 x^{2} )$
	$1 - 16 x + 110 x^{2} - 400 x^{3} + 625 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.295167235301$
Angle rank:	$1$ (numerical)
Jacobians:	$3$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$320$	$368640$	$243863360$	$152510791680$	$95311041953600$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$590$	$15610$	$390430$	$9759850$	$244085870$	$6103206490$	$152586779710$	$3814695203530$	$95367431415950$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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.ak $\times$ 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.ak : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
• 1.25.ag : \(\Q(\sqrt{-1}) \).

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.ae_ak	$2$	2.625.abk_ve
2.25.e_ak	$2$	2.625.abk_ve
2.25.q_eg	$2$	2.625.abk_ve
2.25.ab_u	$3$	(not in LMFDB)