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

• Label: 2.43.au_ha
• 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:	$( 1 - 12 x + 43 x^{2} )( 1 - 8 x + 43 x^{2} )$
	$1 - 20 x + 182 x^{2} - 860 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.132197172840$, $\pm0.291171725172$
Angle rank:	$2$ (numerical)
Jacobians:	$24$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1152$	$3354624$	$6348461184$	$11700338098176$	$21613702474184832$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1814$	$79848$	$3422350$	$147023544$	$6321378278$	$271818590088$	$11688203511454$	$502592665615704$	$21611482752066614$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

• $y^2=26 x^6+5 x^5+8 x^4+9 x^3+31 x^2+33 x+13$
• $y^2=28 x^6+9 x^4+34 x^3+10 x^2+14 x+26$
• $y^2=8 x^5+6 x^4+24 x^3+4 x^2+37 x$
• $y^2=3 x^6+3 x^5+34 x^4+2 x^3+22 x^2+2 x+19$
• $y^2=30 x^6+x^5+5 x^4+19 x^3+5 x^2+x+30$
• $y^2=7 x^6+20 x^5+20 x^4+17 x^3+20 x^2+20 x+7$
• $y^2=17 x^6+5 x^5+31 x^4+21 x^3+31 x^2+5 x+17$
• $y^2=29 x^6+10 x^5+33 x^4+36 x^3+33 x^2+10 x+29$
• $y^2=31 x^6+25 x^5+21 x^4+13 x^3+4 x^2+9 x+38$
• $y^2=27 x^6+30 x^5+27 x^4+29 x^3+33 x^2+5 x+27$
• $y^2=41 x^6+x^5+10 x^4+31 x^3+10 x^2+x+41$
• $y^2=8 x^5+31 x^4+8 x^3+21 x^2+3 x$
• $y^2=24 x^6+2 x^5+8 x^4+26 x^3+20 x^2+34 x+31$
• $y^2=37 x^6+27 x^5+17 x^4+29 x^3+4 x^2+37 x+12$
• $y^2=40 x^6+28 x^5+30 x^4+7 x^3+30 x^2+28 x+40$
• $y^2=7 x^6+3 x^5+26 x^4+34 x^3+26 x^2+3 x+7$
• $y^2=8 x^6+21 x^5+26 x^4+20 x^3+10 x^2+29 x+34$
• $y^2=5 x^6+16 x^5+5 x^4+40 x^3+5 x^2+16 x+5$
• $y^2=26 x^6+22 x^5+38 x^4+36 x^3+38 x^2+22 x+26$
• $y^2=30 x^6+6 x^5+29 x^4+16 x^3+26 x^2+15 x+33$
• $y^2=12 x^6+40 x^5+38 x^4+4 x^3+10 x^2+30 x+42$
• $y^2=42 x^6+21 x^5+25 x^3+21 x+42$
• $y^2=19 x^6+19 x^5+29 x^4+6 x^3+39 x^2+38 x+22$
• $y^2=14 x^5+3 x^4+40 x^3+42 x^2+30 x+18$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.am $\times$ 1.43.ai 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.am : \(\Q(\sqrt{-7}) \).
• 1.43.ai : \(\Q(\sqrt{-3}) \).

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.ae_ak	$2$	(not in LMFDB)
2.43.e_ak	$2$	(not in LMFDB)
2.43.u_ha	$2$	(not in LMFDB)
2.43.ar_fq	$3$	(not in LMFDB)
2.43.b_acs	$3$	(not in LMFDB)