Abelian variety isogeny class 2.5.ag_s over $\F_{5}$

• Label: 2.5.ag_s
• Base field: $\F_{5}$
• 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:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$( 1 - 4 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )$
	$1 - 6 x + 18 x^{2} - 30 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.352416382350$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Isomorphism classes:	6
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$640$	$18056$	$409600$	$9746888$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$26$	$144$	$654$	$3120$	$15626$	$78624$	$392734$	$1956960$	$9765626$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=3 x^6+3 x^5+2 x^4+2 x^3+2 x^2+3 x+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae $\times$ 1.5.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.5.ae : \(\Q(\sqrt{-1}) \).
• 1.5.ac : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{5^{4}}$ is 1.625.o^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{2}}$
  The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag $\times$ 1.25.g. The endomorphism algebra for each factor is:

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

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
2.5.ac_c	$2$	2.25.a_o
2.5.c_c	$2$	2.25.a_o
2.5.g_s	$2$	2.25.a_o