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

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

Invariants

Base field:	$\F_{43}$
Dimension:	$2$
L-polynomial:	$1 - 19 x + 165 x^{2} - 817 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.0635826053657$, $\pm0.344751241674$
Angle rank:	$2$ (numerical)
Number field:	4.0.22725.1
Galois group:	$D_{4}$
Jacobians:	$8$
Cyclic group of points:	no
Non-cyclic primes:	$3$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1179$	$3361329$	$6328814229$	$11684077082541$	$21607248381873264$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$1819$	$79603$	$3417595$	$146979640$	$6321149863$	$271818151441$	$11688205418659$	$502592664546799$	$21611482483233814$

Jacobians and polarizations

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

• $y^2=37 x^6+11 x^5+21 x^4+19 x^3+23 x^2+9 x+21$
• $y^2=20 x^6+7 x^5+39 x^4+17 x^3+18 x^2+18 x+37$
• $y^2=30 x^6+35 x^4+31 x^3+20 x^2+24 x+9$
• $y^2=19 x^6+37 x^5+15 x^4+4 x^3+19 x^2+17 x+2$
• $y^2=9 x^6+x^5+5 x^4+23 x^3+6 x^2+42 x+15$
• $y^2=3 x^6+16 x^5+30 x^4+23 x^3+33 x^2+10 x+27$
• $y^2=10 x^6+23 x^5+22 x^4+39 x^3+41 x^2+30 x+22$
• $y^2=12 x^6+34 x^5+16 x^4+18 x^3+20 x^2+39 x+8$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The endomorphism algebra of this simple isogeny class is 4.0.22725.1.

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.t_gj	$2$	(not in LMFDB)