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

• Label: 2.5.a_ac
• Base field: $\F_{5}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$1 - 2 x^{2} + 25 x^{4}$
Frobenius angles:	$\pm0.217952891576$, $\pm0.782047108424$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{3})\)
Galois group:	$C_2^2$
Jacobians:	$4$
Isomorphism classes:	12

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$24$	$576$	$15768$	$451584$	$9760344$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$22$	$126$	$718$	$3126$	$15910$	$78126$	$388894$	$1953126$	$9755062$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=x^6+x^5+2x^4+x^3+4x^2+3x+1$
• $y^2=2x^6+3x^5+4x^4+3x^3+3x+1$
• $y^2=x^5+3x^4+4x^3+x^2+4x$
• $y^2=x^6+x^3+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{3})\).

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

The base change of $A$ to $\F_{5^{2}}$ is 1.25.ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

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.a_c	$4$	2.625.do_ezm
2.5.ae_i	$8$	(not in LMFDB)
2.5.e_i	$8$	(not in LMFDB)