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

• Label: 2.43.ai_dy
• 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:	$( 1 - 4 x + 43 x^{2} )^{2}$
	$1 - 8 x + 102 x^{2} - 344 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.401344489543$, $\pm0.401344489543$
Angle rank:	$1$ (numerical)
Jacobians:	$68$
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1600$	$3686400$	$6393601600$	$11679989760000$	$21604355049640000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$1990$	$80412$	$3416398$	$146959956$	$6321272470$	$271820333772$	$11688211063198$	$502592581004676$	$21611481725774950$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 68 curves (of which all are hyperelliptic):

• $y^2=36 x^6+42 x^5+30 x^4+42 x^3+33 x^2+19 x+13$
• $y^2=10 x^6+35 x^5+30 x^4+17 x^3+33 x^2+38 x+26$
• $y^2=13 x^5+29 x^4+27 x^3+29 x^2+13 x$
• $y^2=5 x^6+32 x^5+20 x^4+7 x^3+26 x^2+30 x+37$
• $y^2=21 x^6+14 x^5+40 x^4+7 x^3+40 x^2+14 x+21$
• $y^2=13 x^6+20 x^5+x^4+8 x^3+x^2+20 x+13$
• $y^2=16 x^6+11 x^5+41 x^4+x^3+39 x^2+41 x$
• $y^2=38 x^6+33 x^4+33 x^2+38$
• $y^2=18 x^6+20 x^4+20 x^2+18$
• $y^2=4 x^6+7 x^5+25 x^4+16 x^3+16 x^2+36 x+14$
• $y^2=22 x^6+42 x^5+33 x^4+14 x^3+33 x^2+42 x+22$
• $y^2=20 x^6+19 x^5+5 x^4+42 x^3+8 x^2+10 x+2$
• $y^2=26 x^6+19 x^5+11 x^3+19 x+26$
• $y^2=26 x^6+16 x^5+17 x^4+23 x^3+17 x^2+16 x+26$
• $y^2=12 x^6+8 x^5+42 x^4+13 x^3+42 x^2+8 x+12$
• $y^2=20 x^6+5 x^5+19 x^4+13 x^3+19 x^2+5 x+20$
• $y^2=14 x^6+21 x^5+34 x^4+12 x^3+34 x^2+21 x+14$
• $y^2=32 x^6+5 x^5+12 x^4+21 x^3+2 x^2+30 x+32$
• $y^2=29 x^6+35 x^5+12 x^4+31 x^3+2 x^2+38 x+29$
• $y^2=23 x^6+22 x^4+22 x^2+23$
• and

• $y^2=5 x^6+35 x^5+33 x^4+7 x^3+42 x^2+24 x+20$
• $y^2=29 x^6+25 x^5+12 x^4+5 x^3+12 x^2+37 x+38$
• $y^2=37 x^6+10 x^5+24 x^4+8 x^3+24 x^2+10 x+37$
• $y^2=3 x^6+3 x^3+37$
• $y^2=2 x^6+40 x^5+15 x^4+4 x^3+15 x^2+40 x+2$
• $y^2=2 x^6+16 x^5+21 x^4+36 x^3+21 x^2+16 x+2$
• $y^2=25 x^6+19 x^5+10 x^4+27 x^3+35 x^2+7 x+13$
• $y^2=3 x^6+3 x^3+20$
• $y^2=34 x^6+2 x^5+9 x^4+8 x^3+9 x^2+2 x+34$
• $y^2=33 x^6+22 x^5+26 x^4+5 x^3+x^2+10 x+30$
• $y^2=3 x^6+29 x^5+33 x^4+39 x^3+33 x^2+29 x+3$
• $y^2=20 x^6+10 x^4+10 x^2+20$
• $y^2=23 x^6+4 x^5+20 x^4+8 x^3+20 x^2+4 x+23$
• $y^2=9 x^6+22 x^4+22 x^2+9$
• $y^2=39 x^6+6 x^5+12 x^4+20 x^3+39 x^2+15 x+32$
• $y^2=39 x^6+x^4+x^2+39$
• $y^2=7 x^6+31 x^5+22 x^4+2 x^3+22 x^2+31 x+7$
• $y^2=18 x^6+30 x^5+42 x^4+31 x^3+34 x^2+22 x+7$
• $y^2=35 x^6+6 x^5+7 x^4+10 x^3+28 x^2+10 x+4$
• $y^2=40 x^5+27 x^4+15 x^3+27 x^2+40 x$
• $y^2=13 x^6+x^5+27 x^4+39 x^3+27 x^2+x+13$
• $y^2=39 x^6+5 x^5+21 x^4+12 x^3+9 x^2+29 x+22$
• $y^2=18 x^6+21 x^5+39 x^4+11 x^3+39 x^2+21 x+18$
• $y^2=32 x^6+34 x^5+11 x^4+21 x^3+11 x^2+34 x+32$
• $y^2=5 x^6+35 x^5+14 x^4+13 x^3+41 x^2+13 x+5$
• $y^2=35 x^6+6 x^5+3 x^4+42 x^3+32 x^2+9 x+1$
• $y^2=24 x^6+10 x^5+34 x^4+22 x^3+3 x^2+25 x+23$
• $y^2=34 x^6+6 x^5+21 x^4+14 x^3+13 x^2+14 x+30$
• $y^2=28 x^5+41 x^4+3 x^3+41 x^2+28 x$
• $y^2=19 x^6+33 x^5+4 x^4+2 x^3+4 x^2+33 x+19$
• $y^2=24 x^6+33 x^4+33 x^2+24$
• $y^2=3 x^6+39 x^3+12$
• $y^2=22 x^6+25 x^5+9 x^4+41 x^3+23 x^2+21 x+22$
• $y^2=9 x^6+24 x^5+41 x^4+42 x^3+9 x^2+13 x+13$
• $y^2=28 x^6+24 x^5+14 x^4+21 x^3+39 x^2+26 x+20$
• $y^2=20 x^6+16 x^5+23 x^4+x^3+23 x^2+16 x+20$
• $y^2=12 x^6+37 x^5+38 x^4+8 x^3+38 x^2+37 x+12$
• $y^2=27 x^6+16 x^5+42 x^4+23 x^3+42 x^2+16 x+27$
• $y^2=22 x^6+27 x^5+18 x^4+36 x^3+34 x^2+39 x+8$
• $y^2=x^6+9 x^5+18 x^4+41 x^3+40 x^2+x+5$
• $y^2=34 x^6+19 x^5+4 x^4+12 x^3+4 x^2+19 x+34$
• $y^2=22 x^6+22 x^5+30 x^4+x^3+30 x^2+22 x+22$
• $y^2=31 x^6+15 x^5+10 x^4+38 x^3+10 x^2+15 x+31$
• $y^2=19 x^6+42 x^4+42 x^2+19$
• $y^2=21 x^5+10 x^4+2 x^3+4 x^2+24 x$
• $y^2=21 x^6+25 x^5+39 x^4+35 x^3+30 x^2+41 x+4$
• $y^2=13 x^6+4 x^5+13 x^4+33 x^3+13 x^2+4 x+13$
• $y^2=3 x^6+3 x^3+12$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$

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.a_cs	$2$	(not in LMFDB)
2.43.i_dy	$2$	(not in LMFDB)
2.43.e_abb	$3$	(not in LMFDB)