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

• Label: 2.43.ak_eg
• 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:	$( 1 - 6 x + 43 x^{2} )( 1 - 4 x + 43 x^{2} )$
	$1 - 10 x + 110 x^{2} - 430 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.348746511119$, $\pm0.401344489543$
Angle rank:	$2$ (numerical)
Jacobians:	$32$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1520$	$3648000$	$6402077360$	$11688192000000$	$21605448318119600$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$1970$	$80518$	$3418798$	$146967394$	$6321165410$	$271819280998$	$11688211072798$	$502592637117154$	$21611482031104850$

Jacobians and polarizations

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

• $y^2=5 x^6+42 x^5+10 x^4+3 x^3+10 x^2+42 x+5$
• $y^2=29 x^6+41 x^5+14 x^4+33 x^3+10 x^2+6 x+30$
• $y^2=38 x^6+30 x^5+3 x^4+6 x^3+34 x^2+12 x+6$
• $y^2=x^6+34 x^5+23 x^4+37 x^3+23 x^2+34 x+1$
• $y^2=26 x^6+14 x^5+40 x^4+36 x^3+17 x^2+10 x+34$
• $y^2=9 x^6+22 x^5+15 x^4+34 x^3+25 x^2+42 x+13$
• $y^2=35 x^6+14 x^5+28 x^4+x^3+28 x^2+14 x+35$
• $y^2=28 x^6+4 x^5+5 x^4+32 x^3+5 x^2+4 x+28$
• $y^2=x^6+31 x^5+40 x^4+19 x^3+40 x^2+31 x+1$
• $y^2=5 x^6+39 x^5+10 x^4+41 x^2+5 x+12$
• $y^2=36 x^6+6 x^5+13 x^4+2 x^3+13 x^2+6 x+36$
• $y^2=34 x^6+24 x^5+7 x^4+5 x^3+7 x^2+24 x+34$
• $y^2=6 x^6+9 x^5+9 x^4+35 x^3+38 x^2+41 x+31$
• $y^2=20 x^6+3 x^5+12 x^4+23 x^3+12 x^2+3 x+20$
• $y^2=23 x^6+16 x^5+37 x^4+16 x^3+18 x^2+15 x+24$
• $y^2=17 x^6+7 x^4+34 x^3+7 x^2+17$
• $y^2=20 x^6+37 x^5+27 x^4+19 x^3+3 x^2+27 x+33$
• $y^2=23 x^6+10 x^5+40 x^4+9 x^3+40 x^2+10 x+23$
• $y^2=33 x^6+20 x^5+13 x^4+25 x^3+40 x^2+26 x+3$
• $y^2=22 x^6+37 x^5+3 x^4+8 x^3+32 x^2+34 x+8$
• and

• $y^2=34 x^6+37 x^5+20 x^4+17 x^3+20 x^2+37 x+34$
• $y^2=17 x^6+37 x^5+2 x^4+21 x^3+42 x^2+20 x+14$
• $y^2=30 x^6+35 x^5+41 x^4+33 x^3+41 x^2+35 x+30$
• $y^2=19 x^6+10 x^5+4 x^4+7 x^3+11 x^2+38 x+33$
• $y^2=30 x^6+8 x^5+30 x^4+24 x^3+28 x^2+2 x+7$
• $y^2=38 x^5+13 x^4+7 x^3+25 x^2+24 x$
• $y^2=18 x^6+29 x^5+16 x^4+37 x^3+16 x^2+29 x+18$
• $y^2=5 x^6+10 x^5+16 x^4+x^3+16 x^2+10 x+5$
• $y^2=3 x^6+5 x^5+23 x^4+39 x^3+23 x^2+5 x+3$
• $y^2=3 x^6+18 x^5+27 x^4+4 x^3+27 x^2+18 x+3$
• $y^2=42 x^6+19 x^5+15 x^4+19 x^3+15 x^2+19 x+42$
• $y^2=19 x^6+19 x^5+22 x^4+15 x^3+22 x^2+19 x+19$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.ag $\times$ 1.43.ae 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.ag : \(\Q(\sqrt{-34}) \).
• 1.43.ae : \(\Q(\sqrt{-39}) \).

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.ac_ck	$2$	(not in LMFDB)
2.43.c_ck	$2$	(not in LMFDB)
2.43.k_eg	$2$	(not in LMFDB)