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

• Label: 2.43.s_gk
• 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:	$( 1 + 8 x + 43 x^{2} )( 1 + 10 x + 43 x^{2} )$
	$1 + 18 x + 166 x^{2} + 774 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.708828274828$, $\pm0.776024765496$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2808$	$3436992$	$6257271384$	$11711179044864$	$21607674870172248$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$1858$	$78698$	$3425518$	$146982542$	$6321326578$	$271819866842$	$11688191359006$	$502592640141854$	$21611482381579618$

Jacobians and polarizations

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

• $y^2=39 x^6+26 x^5+14 x^4+15 x^3+14 x^2+26 x+39$
• $y^2=8 x^6+9 x^5+37 x^4+20 x^3+37 x^2+9 x+8$
• $y^2=11 x^6+9 x^5+42 x^4+26 x^3+42 x^2+9 x+11$
• $y^2=31 x^6+42 x^5+39 x^4+42 x^3+39 x^2+42 x+31$
• $y^2=37 x^6+24 x^5+26 x^4+35 x^3+12 x^2+13 x+20$
• $y^2=3 x^6+15 x^5+13 x^4+32 x^3+13 x^2+15 x+3$
• $y^2=25 x^6+9 x^5+21 x^4+8 x^3+x^2+36 x+15$
• $y^2=26 x^6+2 x^5+19 x^4+7 x^3+34 x^2+7 x+18$
• $y^2=36 x^6+25 x^5+22 x^4+38 x^3+7 x^2+41 x+13$
• $y^2=38 x^6+37 x^5+16 x^4+16 x^2+37 x+38$
• $y^2=32 x^6+20 x^5+22 x^4+9 x^3+22 x^2+20 x+32$
• $y^2=2 x^6+13 x^5+32 x^4+11 x^3+32 x^2+13 x+2$
• $y^2=18 x^6+18 x^5+12 x^4+36 x^3+12 x^2+18 x+18$
• $y^2=23 x^6+9 x^5+35 x^4+19 x^3+35 x^2+9 x+23$
• $y^2=25 x^6+6 x^5+x^4+9 x^3+23 x^2+35 x+36$
• $y^2=28 x^6+27 x^5+32 x^4+19 x^3+32 x^2+27 x+28$
• $y^2=38 x^6+26 x^5+22 x^4+7 x^3+5 x^2+20 x+31$
• $y^2=21 x^6+19 x^5+9 x^4+41 x^2+2 x+41$
• $y^2=7 x^6+2 x^5+27 x^4+27 x^2+2 x+7$
• $y^2=15 x^6+24 x^5+24 x^4+6 x^3+24 x^2+24 x+15$
• $y^2=5 x^6+25 x^5+27 x^4+29 x^3+26 x^2+21 x+5$
• $y^2=33 x^6+16 x^5+x^4+3 x^3+x^2+16 x+33$
• $y^2=38 x^6+33 x^5+41 x^4+31 x^3+15 x^2+18 x+40$
• $y^2=15 x^6+33 x^5+34 x^4+41 x^3+5 x^2+7 x+13$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.i $\times$ 1.43.k 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.i : \(\Q(\sqrt{-3}) \).
• 1.43.k : \(\Q(\sqrt{-2}) \).

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.as_gk	$2$	(not in LMFDB)
2.43.ac_g	$2$	(not in LMFDB)
2.43.c_g	$2$	(not in LMFDB)
2.43.ad_abs	$3$	(not in LMFDB)
2.43.p_fg	$3$	(not in LMFDB)