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

• Label: 2.43.au_gz
• Base field: $\F_{43}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{43}$
Dimension:	$2$
L-polynomial:	$1 - 20 x + 181 x^{2} - 860 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.117188704887$, $\pm0.298340802765$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-202 -14 \sqrt{5}})\)
Galois group:	$D_{4}$
Jacobians:	$8$
Cyclic group of points:	yes

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1151$	$3350561$	$6343649024$	$11697347891321$	$21612526371340911$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1812$	$79788$	$3421476$	$147015544$	$6321332118$	$271818489288$	$11688204601668$	$502592680738164$	$21611482859082772$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=18 x^6+2 x^5+9 x^4+20 x^3+x^2+26 x+4$
• $y^2=42 x^6+31 x^5+13 x^4+21 x^3+16 x^2+38 x+8$
• $y^2=26 x^6+41 x^5+9 x^4+20 x^3+34 x^2+25 x+39$
• $y^2=7 x^6+28 x^5+x^3+23 x^2+31 x+26$
• $y^2=26 x^6+10 x^5+31 x^4+29 x^3+12 x^2+22 x+18$
• $y^2=25 x^6+27 x^5+10 x^4+10 x^3+25 x^2+19 x+33$
• $y^2=33 x^6+12 x^5+15 x^4+12 x^3+34 x^2+28 x+2$
• $y^2=9 x^6+24 x^5+10 x^4+3 x^3+9 x^2+37 x+20$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-202 -14 \sqrt{5}})\).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.43.u_gz	$2$	(not in LMFDB)