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

• Label: 2.43.ab_abq
• 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:	$1 - x - 42 x^{2} - 43 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.142372185324$, $\pm0.809038851991$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-19})\)
Galois group:	$C_2^2$
Jacobians:	$47$
Isomorphism classes:	102
Cyclic group of points:	no
Non-cyclic primes:	$2, 3, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1764$	$3266928$	$6301184400$	$11700267413184$	$21610154614577004$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$43$	$1765$	$79252$	$3422329$	$146999413$	$6321648310$	$271819142071$	$11688205879729$	$502592668803916$	$21611482100826325$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 47 curves (of which all are hyperelliptic):

• $y^2=x^6+x^3+14$
• $y^2=23 x^6+5 x^5+26 x^4+25 x^3+23 x^2+41 x+36$
• $y^2=20 x^6+21 x^5+16 x^4+36 x^3+9 x^2+7 x$
• $y^2=6 x^6+7 x^5+36 x^4+23 x^3+31 x^2+41 x+26$
• $y^2=21 x^6+21 x^5+12 x^4+32 x^3+16 x^2+42 x+5$
• $y^2=21 x^6+6 x^5+28 x^4+x^3+6 x^2+24 x+42$
• $y^2=26 x^6+8 x^5+6 x^4+37 x^3+38 x^2+34 x+33$
• $y^2=19 x^6+8 x^5+31 x^4+31 x^3+4 x^2+2 x$
• $y^2=31 x^6+16 x^5+26 x^4+10 x^3+38 x^2+33 x+10$
• $y^2=31 x^6+9 x^5+x^4+3 x^3+16 x^2+42 x+9$
• $y^2=16 x^6+25 x^5+15 x^4+19 x^3+12 x^2+12 x$
• $y^2=29 x^6+35 x^5+19 x^4+30 x^3+36 x^2+22 x+20$
• $y^2=32 x^6+20 x^5+9 x^4+23 x^3+12 x^2+10 x+12$
• $y^2=41 x^6+34 x^5+42 x^4+19 x^3+13 x^2+12 x+42$
• $y^2=2 x^5+9 x^4+18 x^3+17 x^2+5 x+9$
• $y^2=39 x^6+40 x^5+25 x^4+21 x^3+39 x^2+31 x+11$
• $y^2=2 x^6+35 x^5+10 x^4+2 x^3+18 x^2+11 x+32$
• $y^2=12 x^6+7 x^5+32 x^4+30 x^3+20 x^2+33 x+18$
• $y^2=36 x^6+27 x^5+4 x^4+34 x^3+22 x^2+28 x+9$
• $y^2=2 x^6+17 x^5+33 x^4+42 x^3+29 x^2+15 x+19$
• and

• $y^2=12 x^6+27 x^5+41 x^4+2 x^3+14 x^2+41 x+38$
• $y^2=24 x^6+18 x^5+39 x^4+34 x^3+28 x^2+23 x+8$
• $y^2=6 x^6+8 x^5+21 x^4+32 x^3+25 x^2+18 x$
• $y^2=11 x^6+2 x^5+x^4+28 x^3+28 x^2+29 x+37$
• $y^2=37 x^6+37 x^5+20 x^4+30 x^3+17 x^2+23 x+31$
• $y^2=11 x^6+23 x^5+37 x^4+20 x^3+2 x^2+12 x+16$
• $y^2=17 x^6+31 x^5+27 x^4+24 x^3+11 x^2+9 x+24$
• $y^2=10 x^6+7 x^5+33 x^4+23 x^3+34 x^2+2 x+32$
• $y^2=23 x^6+9 x^4+23 x^3+2 x^2+8 x+38$
• $y^2=x^6+x^3+40$
• $y^2=5 x^6+17 x^5+12 x^4+36 x^3+11 x^2+16 x+23$
• $y^2=21 x^6+9 x^5+39 x^4+2 x^3+11 x^2+34 x+7$
• $y^2=22 x^6+13 x^5+21 x^4+x^3+13 x^2+32 x+20$
• $y^2=x^6+3 x^3+9$
• $y^2=x^6+x^3+10$
• $y^2=18 x^6+11 x^5+22 x^4+41 x^3+33 x^2+30 x+16$
• $y^2=21 x^6+21 x^5+26 x^4+35 x^3+20 x^2+24 x+6$
• $y^2=x^6+2 x^5+32 x^4+40 x^3+10 x^2+35 x+17$
• $y^2=22 x^6+23 x^5+39 x^4+5 x^3+6 x^2+32 x+36$
• $y^2=3 x^6+12 x^5+33 x^4+38 x^3+31 x^2+32 x+20$
• $y^2=17 x^6+20 x^5+34 x^4+23 x^3+x^2+30$
• $y^2=40 x^6+10 x^5+9 x^4+17 x^3+35 x^2+20 x+20$
• $y^2=23 x^6+13 x^5+11 x^4+9 x^3+26 x^2+23 x+36$
• $y^2=37 x^6+5 x^5+8 x^4+9 x^3+31 x^2+10 x+2$
• $y^2=33 x^5+9 x^4+36 x^3+x^2+11 x+11$
• $y^2=4 x^6+30 x^4+29 x^3+3 x^2+11 x+2$
• $y^2=30 x^6+11 x^5+33 x^4+10 x^3+40 x^2+34 x+37$

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 \(\Q(\sqrt{-3}, \sqrt{-19})\).

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

The base change of $A$ to $\F_{43^{3}}$ is 1.79507.aey^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

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.b_abq	$2$	(not in LMFDB)
2.43.c_dj	$3$	(not in LMFDB)
2.43.ac_dj	$6$	(not in LMFDB)