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

• Label: 2.43.aj_bm
• Base field: $\F_{43}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{43}$
Dimension:	$2$
L-polynomial:	$1 - 9 x + 38 x^{2} - 387 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.0740749180724$, $\pm0.592591748594$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-91})\)
Galois group:	$C_2^2$
Jacobians:	$38$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1492$	$3407728$	$6253013776$	$11675653089984$	$21613611205252732$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$1845$	$78644$	$3415129$	$147022925$	$6321307830$	$271817739935$	$11688206930929$	$502592656775852$	$21611482228966725$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 38 curves (of which all are hyperelliptic):

• $y^2=39 x^6+2 x^5+40 x^3+36 x^2+14 x+13$
• $y^2=27 x^6+23 x^5+38 x^4+40 x^3+26 x^2+2 x+19$
• $y^2=19 x^6+32 x^5+3 x^4+35 x^2+38 x+7$
• $y^2=5 x^6+28 x^5+35 x^4+40 x^3+42 x^2+40 x+22$
• $y^2=19 x^6+35 x^5+7 x^4+41 x^3+31 x^2+16 x+12$
• $y^2=11 x^6+34 x^5+8 x^4+41 x^3+36 x^2+23 x+3$
• $y^2=33 x^6+31 x^5+17 x^4+27 x^3+x+11$
• $y^2=27 x^6+27 x^5+40 x^4+30 x^3+22 x^2+7 x+7$
• $y^2=31 x^6+21 x^5+6 x^4+28 x^3+38 x^2+14 x+1$
• $y^2=13 x^6+24 x^5+31 x^4+35 x^3+38 x^2+27 x+29$
• $y^2=34 x^6+18 x^5+11 x^4+40 x^3+5 x^2+27 x+2$
• $y^2=15 x^6+24 x^5+9 x^4+35 x^3+40 x^2+32 x+3$
• $y^2=33 x^6+42 x^5+7 x^4+38 x^3+31 x^2+29 x+11$
• $y^2=18 x^6+6 x^5+27 x^4+6 x^3+38 x^2+34 x+39$
• $y^2=10 x^6+18 x^5+29 x^4+34 x^3+17 x^2+x+19$
• $y^2=36 x^6+30 x^5+22 x^4+40 x^3+19 x^2+x+36$
• $y^2=29 x^6+26 x^5+41 x^4+18 x^3+25 x^2+30 x+31$
• $y^2=22 x^6+5 x^5+x^4+26 x^3+42 x^2+19 x+2$
• $y^2=22 x^6+9 x^5+29 x^4+24 x^2+11 x+5$
• $y^2=7 x^6+17 x^5+7 x^4+26 x^3+25 x^2+40 x+3$
• and

• $y^2=41 x^6+14 x^5+6 x^4+x^3+18 x^2+32 x+24$
• $y^2=29 x^6+x^5+40 x^4+30 x^3+21 x^2+2 x+4$
• $y^2=12 x^6+39 x^5+22 x^4+40 x^3+32 x^2+5 x+11$
• $y^2=14 x^6+x^5+30 x^4+42 x^3+35 x^2+37 x+29$
• $y^2=27 x^6+42 x^5+39 x^4+29 x^3+18 x^2+42 x+8$
• $y^2=22 x^6+33 x^5+19 x^4+27 x^3+20 x^2+21 x+25$
• $y^2=27 x^6+28 x^5+5 x^4+14 x^3+12 x^2+9 x+18$
• $y^2=12 x^6+28 x^5+38 x^4+29 x^3+18 x^2+27 x+37$
• $y^2=42 x^6+17 x^5+40 x^4+41 x^3+14 x^2+15 x+14$
• $y^2=35 x^6+34 x^5+5 x^4+31 x^3+10 x^2+29 x+8$
• $y^2=42 x^6+3 x^5+12 x^4+36 x^3+8 x^2+2 x+11$
• $y^2=16 x^6+21 x^5+26 x^4+38 x^3+28 x^2+33 x+16$
• $y^2=5 x^6+27 x^5+5 x^4+2 x^3+19 x^2+34 x+7$
• $y^2=29 x^6+16 x^5+36 x^4+30 x^3+38 x^2+12 x+16$
• $y^2=39 x^6+8 x^5+13 x^4+14 x^3+33 x^2+4 x+17$
• $y^2=22 x^6+26 x^5+25 x^4+39 x^3+2 x^2+37 x+33$
• $y^2=21 x^6+30 x^5+38 x^4+22 x^3+30 x^2+35 x+6$
• $y^2=35 x^6+40 x^5+x^4+10 x^3+19 x^2+21 x+11$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{43^{3}}$.

Endomorphism algebra over $\F_{43}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-91})\).

Endomorphism algebra over $\overline{\F}_{43}$

The base change of $A$ to $\F_{43^{3}}$ is 1.79507.aqq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-91}) \)$)$

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.j_bm	$2$	(not in LMFDB)
2.43.s_gl	$3$	(not in LMFDB)
2.43.as_gl	$6$	(not in LMFDB)