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

• Label: 2.43.at_gs
• 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:	$( 1 - 11 x + 43 x^{2} )( 1 - 8 x + 43 x^{2} )$
	$1 - 19 x + 174 x^{2} - 817 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.183291501244$, $\pm0.291171725172$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
Isomorphism classes:	56
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1188$	$3397680$	$6369908688$	$11707657790400$	$21615418248855948$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$1837$	$80116$	$3424489$	$147035215$	$6321402934$	$271818232837$	$11688197507281$	$502592608205548$	$21611482341632557$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=26 x^6+19 x^5+19 x^4+20 x^3+17 x^2+35 x+19$
• $y^2=7 x^6+x^5+3 x^4+26 x^3+6 x^2+41 x+14$
• $y^2=7 x^6+14 x^5+12 x^4+4 x^3+26 x^2+40 x+33$
• $y^2=27 x^6+20 x^5+29 x^4+20 x^3+12 x^2+16 x+27$
• $y^2=2 x^6+35 x^5+32 x^4+16 x^3+16 x^2+24 x+40$
• $y^2=17 x^6+13 x^5+22 x^4+39 x^3+18 x^2+5 x+27$
• $y^2=22 x^6+14 x^5+27 x^4+13 x^3+24 x^2+7 x+39$
• $y^2=34 x^6+11 x^5+19 x^4+27 x^3+32 x^2+17 x+20$
• $y^2=30 x^6+10 x^5+24 x^3+8 x^2+41 x+23$
• $y^2=5 x^6+2 x^5+41 x^4+35 x^3+12 x^2+32 x+29$
• $y^2=32 x^6+26 x^5+33 x^4+4 x^3+17 x^2+38 x+18$
• $y^2=3 x^6+8 x^5+5 x^4+22 x^3+36 x^2+6 x+22$

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.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.al : \(\Q(\sqrt{-51}) \).
• 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.ad_ac	$2$	(not in LMFDB)
2.43.d_ac	$2$	(not in LMFDB)
2.43.t_gs	$2$	(not in LMFDB)
2.43.aq_fl	$3$	(not in LMFDB)
2.43.c_acf	$3$	(not in LMFDB)