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

• Label: 2.43.ah_g
• 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:	$1 - 7 x + 6 x^{2} - 301 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.0125491117519$, $\pm0.654117554915$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{41})\)
Galois group:	$C_2^2$
Jacobians:	$8$
Isomorphism classes:	8
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1548$	$3349872$	$6232786704$	$11680239893184$	$21610338942088548$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$1813$	$78388$	$3416473$	$147000667$	$6321053878$	$271817863417$	$11688198864241$	$502592527848364$	$21611482079749093$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=2 x^6+34 x^5+19 x^4+8 x^3+38 x^2+4 x+11$
• $y^2=x^6+x^3+31$
• $y^2=27 x^6+8 x^5+x^4+28 x^3+42 x^2+27 x+42$
• $y^2=26 x^6+x^5+39 x^4+9 x^3+25 x^2+37 x+18$
• $y^2=42 x^6+23 x^5+14 x^3+31 x^2+3 x+22$
• $y^2=29 x^6+5 x^5+8 x^4+30 x^3+37 x^2+18 x+20$
• $y^2=42 x^6+31 x^5+25 x^4+3 x^3+13 x^2+19 x+28$
• $y^2=19 x^6+22 x^5+13 x^4+15 x^3+7 x^2+6 x+26$

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 \(\Q(\sqrt{-3}, \sqrt{41})\).

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

The base change of $A$ to $\F_{43^{3}}$ is 1.79507.avo^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-123}) \)$)$

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.h_g	$2$	(not in LMFDB)
2.43.o_ff	$3$	(not in LMFDB)
2.43.ao_ff	$6$	(not in LMFDB)