Abelian variety isogeny class 2.43.ax_ii over $\F_{43}$

• Label: 2.43.ax_ii
• Base field: $\F_{43}$
• 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:	$\F_{43}$
Dimension:	$2$
L-polynomial:	$( 1 - 13 x + 43 x^{2} )( 1 - 10 x + 43 x^{2} )$
	$1 - 23 x + 216 x^{2} - 989 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.0421616081610$, $\pm0.223975234504$
Angle rank:	$2$ (numerical)
Jacobians:	$1$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1054$	$3244212$	$6303084424$	$11689259187744$	$21611985974015074$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$21$	$1753$	$79278$	$3419113$	$147011871$	$6321326578$	$271817764869$	$11688191506321$	$502592550577674$	$21611482027126393$

Jacobians and polarizations

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

• $y^2=16 x^6+40 x^5+15 x^4+21 x^3+9 x^2+24 x+41$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{43}$.

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.an $\times$ 1.43.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.43.an : \(\Q(\sqrt{-3}) \).
• 1.43.ak : \(\Q(\sqrt{-2}) \).

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
2.43.ad_abs	$2$	(not in LMFDB)
2.43.d_abs	$2$	(not in LMFDB)
2.43.x_ii	$2$	(not in LMFDB)
2.43.af_bk	$3$	(not in LMFDB)
2.43.ac_g	$3$	(not in LMFDB)