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

• Label: 2.43.at_gr
• 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:	$1 - 19 x + 173 x^{2} - 817 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.169319344707$, $\pm0.300345078312$
Angle rank:	$2$ (numerical)
Number field:	4.0.843141.1
Galois group:	$D_{4}$
Jacobians:	$6$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1187$	$3393633$	$6365338541$	$11705091570189$	$21614622116806832$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$1835$	$80059$	$3423739$	$147029800$	$6321392615$	$271818492985$	$11688201237859$	$502592635284367$	$21611482439590550$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=14 x^6+4 x^5+9 x^4+11 x^3+7 x^2+7 x+25$
• $y^2=19 x^6+40 x^5+7 x^4+18 x^3+29 x^2+32 x+21$
• $y^2=x^6+2 x^4+12 x^3+5 x^2+35 x+33$
• $y^2=13 x^6+20 x^5+19 x^4+9 x^3+34 x^2+9 x+19$
• $y^2=16 x^6+16 x^5+12 x^4+8 x^3+12 x^2+19 x+33$
• $y^2=20 x^6+7 x^5+6 x^4+30 x^3+3 x^2+28 x+18$

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 4.0.843141.1.

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.t_gr	$2$	(not in LMFDB)