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

• Label: 2.43.ar_fw
• 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:	$( 1 - 11 x + 43 x^{2} )( 1 - 6 x + 43 x^{2} )$
	$1 - 17 x + 152 x^{2} - 731 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.183291501244$, $\pm0.348746511119$
Angle rank:	$2$ (numerical)
Jacobians:	$16$
Isomorphism classes:	104
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1254$	$3448500$	$6372933336$	$11700760500000$	$21612454380561594$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$27$	$1865$	$80154$	$3422473$	$147015057$	$6321361970$	$271819076619$	$11688206401873$	$502592632274862$	$21611482070426825$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

• $y^2=30 x^6+8 x^5+38 x^4+30 x^3+17 x^2+18 x+22$
• $y^2=18 x^6+36 x^4+3 x^3+7 x^2+29 x+32$
• $y^2=2 x^6+19 x^5+42 x^4+36 x^3+27 x^2+x+12$
• $y^2=19 x^6+10 x^5+20 x^4+36 x^3+2 x^2+16 x+26$
• $y^2=29 x^6+27 x^5+3 x^4+39 x^3+33 x^2+24 x+17$
• $y^2=32 x^6+18 x^5+10 x^4+20 x^3+x^2+15 x+17$
• $y^2=28 x^6+30 x^5+37 x^4+32 x^3+13 x^2+14 x+8$
• $y^2=39 x^6+16 x^5+14 x^4+5 x^3+2 x^2+2 x+8$
• $y^2=29 x^6+x^5+42 x^4+21 x^3+20 x^2+7 x$
• $y^2=28 x^6+27 x^5+14 x^4+4 x^3+36 x^2+21 x+29$
• $y^2=42 x^6+32 x^5+34 x^4+38 x^3+31 x^2+34 x+6$
• $y^2=42 x^6+15 x^5+x^4+13 x^3+11 x^2+24 x+19$
• $y^2=8 x^6+26 x^5+17 x^4+25 x^3+3 x^2+36 x+14$
• $y^2=10 x^6+11 x^5+40 x^4+22 x^3+9 x^2+24 x+12$
• $y^2=8 x^6+24 x^5+10 x^4+19 x^3+29 x^2+40 x+7$
• $y^2=22 x^6+16 x^5+25 x^4+17 x^3+26 x^2+40 x+34$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.al $\times$ 1.43.ag 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.al : \(\Q(\sqrt{-51}) \).
• 1.43.ag : \(\Q(\sqrt{-34}) \).

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.af_u	$2$	(not in LMFDB)
2.43.f_u	$2$	(not in LMFDB)
2.43.r_fw	$2$	(not in LMFDB)