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

• Label: 2.43.ao_fe
• 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:	$( 1 - 8 x + 43 x^{2} )( 1 - 6 x + 43 x^{2} )$
	$1 - 14 x + 134 x^{2} - 602 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.291171725172$, $\pm0.348746511119$
Angle rank:	$2$ (numerical)
Jacobians:	$32$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1368$	$3556800$	$6407521848$	$11703294720000$	$21609504699361848$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$1922$	$80586$	$3423214$	$146994990$	$6321099314$	$271817384442$	$11688202187806$	$502592669160318$	$21611482607640482$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

• $y^2=13 x^6+26 x^5+39 x^3+27 x+13$
• $y^2=8 x^6+41 x^5+39 x^4+x^3+37 x^2+17 x+27$
• $y^2=42 x^6+31 x^5+20 x^4+42 x^3+20 x^2+31 x+42$
• $y^2=17 x^6+25 x^5+25 x^4+14 x^3+11 x^2+10 x+14$
• $y^2=26 x^6+10 x^5+34 x^4+39 x^3+2 x^2+35 x+18$
• $y^2=33 x^6+14 x^5+20 x^4+18 x^3+27 x^2+21 x+12$
• $y^2=7 x^6+14 x^5+27 x^4+14 x^3+27 x^2+14 x+7$
• $y^2=34 x^6+41 x^5+41 x^4+10 x^3+41 x^2+41 x+34$
• $y^2=28 x^6+36 x^5+18 x^4+7 x^3+18 x^2+36 x+28$
• $y^2=9 x^6+11 x^5+12 x^4+9 x^3+18 x^2+14 x+25$
• $y^2=5 x^6+33 x^5+31 x^3+33 x+5$
• $y^2=22 x^6+32 x^5+38 x^4+16 x^3+15 x^2+30 x+8$
• $y^2=2 x^5+20 x^4+24 x^3+18 x^2+30 x$
• $y^2=37 x^6+20 x^5+8 x^4+29 x^3+8 x^2+20 x+37$
• $y^2=41 x^6+33 x^5+14 x^4+6 x^3+35 x^2+2 x+1$
• $y^2=10 x^6+9 x^5+12 x^4+15 x^3+12 x^2+9 x+10$
• $y^2=32 x^6+10 x^5+17 x^4+10 x^3+17 x^2+10 x+32$
• $y^2=36 x^6+11 x^5+35 x^4+22 x^3+35 x^2+11 x+36$
• $y^2=18 x^6+33 x^5+35 x^4+6 x^3+35 x^2+33 x+18$
• $y^2=21 x^6+20 x^5+5 x^4+12 x^3+3 x^2+33 x+41$
• and

• $y^2=18 x^6+26 x^5+20 x^4+39 x^3+20 x^2+26 x+18$
• $y^2=12 x^6+37 x^5+5 x^4+15 x^3+5 x^2+37 x+12$
• $y^2=27 x^6+28 x^5+13 x^4+5 x^3+14 x^2+34 x+32$
• $y^2=9 x^6+42 x^5+5 x^4+28 x^3+5 x^2+42 x+9$
• $y^2=32 x^6+27 x^5+3 x^4+3 x^3+34 x^2+28 x+39$
• $y^2=34 x^6+42 x^5+34 x^4+37 x^3+34 x^2+42 x+34$
• $y^2=42 x^6+24 x^5+13 x^4+10 x^3+13 x^2+24 x+42$
• $y^2=42 x^6+24 x^5+13 x^4+10 x^2+17 x+8$
• $y^2=2 x^6+22 x^5+5 x^4+18 x^3+12 x^2+39 x+39$
• $y^2=19 x^6+8 x^5+28 x^4+7 x^3+42 x^2+18 x+5$
• $y^2=33 x^6+36 x^5+10 x^4+17 x^3+10 x^2+36 x+33$
• $y^2=27 x^6+41 x^4+4 x^3+15 x^2+42$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.ai $\times$ 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.ai : \(\Q(\sqrt{-3}) \).
• 1.43.ag : \(\Q(\sqrt{-34}) \).

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.ac_bm	$2$	(not in LMFDB)
2.43.c_bm	$2$	(not in LMFDB)
2.43.o_fe	$2$	(not in LMFDB)
2.43.al_em	$3$	(not in LMFDB)
2.43.h_i	$3$	(not in LMFDB)