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

• Label: 2.43.as_gc
• 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:	$( 1 - 12 x + 43 x^{2} )( 1 - 6 x + 43 x^{2} )$
	$1 - 18 x + 158 x^{2} - 774 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.132197172840$, $\pm0.348746511119$
Angle rank:	$2$ (numerical)
Jacobians:	$40$
Isomorphism classes:	160
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1216$	$3404800$	$6351475648$	$11693445120000$	$21610738841154496$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$26$	$1842$	$79886$	$3420334$	$147003386$	$6321337314$	$271819433870$	$11688212406046$	$502592689685018$	$21611482480860882$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=9 x^6+25 x^5+28 x^4+6 x^3+34 x^2+13 x+22$
• $y^2=9 x^6+5 x^5+31 x^4+33 x^3+36 x^2+3 x+20$
• $y^2=7 x^6+4 x^5+12 x^4+10 x^3+16 x^2+37 x+28$
• $y^2=38 x^6+38 x^5+35 x^4+31 x^3+21 x^2+9 x+3$
• $y^2=17 x^6+21 x^5+5 x^4+29 x^3+27 x^2+28 x+29$
• $y^2=28 x^6+31 x^5+19 x^4+11 x^3+2 x^2+25 x+18$
• $y^2=18 x^6+15 x^5+21 x^4+5 x^3+2 x^2+14 x+18$
• $y^2=34 x^6+23 x^5+13 x^4+5 x^3+38 x^2+32 x+3$
• $y^2=10 x^6+14 x^5+25 x^4+24 x^3+11 x^2+7 x+17$
• $y^2=20 x^6+26 x^5+27 x^4+2 x^3+27 x^2+26 x+20$
• $y^2=12 x^6+14 x^5+25 x^4+5 x^3+17 x^2+15 x$
• $y^2=7 x^6+29 x^5+27 x^4+x^3+18 x^2+28 x+9$
• $y^2=9 x^6+17 x^5+33 x^4+14 x^3+28 x^2+32 x+38$
• $y^2=33 x^6+29 x^5+28 x^4+4 x^3+17 x^2+23 x$
• $y^2=5 x^6+31 x^5+9 x^4+18 x^3+12 x^2+x+8$
• $y^2=22 x^6+12 x^5+11 x^4+13 x^3+12 x^2+42 x+7$
• $y^2=25 x^6+18 x^5+x^4+12 x^3+28 x^2+32 x+30$
• $y^2=34 x^5+40 x^4+36 x^3+32 x^2+13 x+7$
• $y^2=37 x^6+38 x^5+22 x^4+23 x^3+19 x^2+4 x+19$
• $y^2=5 x^6+7 x^5+21 x^4+35 x^3+18 x^2+36 x+37$
• and

• $y^2=8 x^6+3 x^5+31 x^4+x^3+9 x^2+38 x+14$
• $y^2=2 x^6+41 x^5+30 x^4+5 x^3+31 x^2+18 x+20$
• $y^2=19 x^6+15 x^5+3 x^4+17 x^3+21 x^2+41 x+3$
• $y^2=22 x^6+26 x^5+36 x^4+25 x^3+36 x^2+26 x+22$
• $y^2=9 x^6+35 x^5+36 x^4+14 x^3+41 x^2+9 x+14$
• $y^2=19 x^6+37 x^5+11 x^4+32 x^3+25 x^2+28 x+27$
• $y^2=34 x^6+38 x^5+42 x^4+29 x^3+21 x^2+15 x+15$
• $y^2=20 x^6+2 x^5+22 x^4+20 x^3+33 x^2+26 x+3$
• $y^2=2 x^6+35 x^5+3 x^4+23 x^3+42 x^2+36 x+18$
• $y^2=26 x^5+32 x^4+16 x^3+2 x^2+10 x+13$
• $y^2=21 x^6+23 x^5+5 x^4+17 x^3+31 x^2+6 x+42$
• $y^2=7 x^6+28 x^5+3 x^4+38 x^3+10 x^2+40 x+32$
• $y^2=3 x^6+27 x^5+34 x^4+7 x^3+23 x^2+25 x+42$
• $y^2=37 x^6+31 x^5+5 x^4+22 x^3+30 x^2+27 x+7$
• $y^2=27 x^6+29 x^5+22 x^4+32 x^3+22 x^2+29 x+27$
• $y^2=12 x^6+10 x^5+33 x^4+8 x^3+19 x^2+15 x+32$
• $y^2=2 x^6+31 x^5+36 x^4+42 x^3+33 x^2+16 x+5$
• $y^2=x^5+37 x^4+6 x^3+9 x^2+31 x+26$
• $y^2=31 x^6+23 x^5+7 x^4+30 x^3+21 x^2+16 x+37$
• $y^2=40 x^6+19 x^5+3 x^4+16 x^3+3 x^2+19 x+40$

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.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.am : \(\Q(\sqrt{-7}) \).
• 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.ag_o	$2$	(not in LMFDB)
2.43.g_o	$2$	(not in LMFDB)
2.43.s_gc	$2$	(not in LMFDB)