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

• Label: 2.43.aw_hy
• 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:	$( 1 - 12 x + 43 x^{2} )( 1 - 10 x + 43 x^{2} )$
	$1 - 22 x + 206 x^{2} - 946 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.132197172840$, $\pm0.223975234504$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1088$	$3290112$	$6330215744$	$11701322809344$	$21616524967830848$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$1778$	$79618$	$3422638$	$147042742$	$6321564578$	$271819404802$	$11688201577246$	$502592604256534$	$21611482254800018$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=3 x^6+35 x^5+42 x^4+16 x^3+33 x^2+17 x+33$
• $y^2=21 x^6+21 x^5+41 x^4+18 x^3+10 x^2+9 x+41$
• $y^2=38 x^6+8 x^5+15 x^4+5 x^3+15 x^2+8 x+38$
• $y^2=18 x^6+17 x^5+28 x^4+12 x^3+28 x^2+17 x+18$
• $y^2=33 x^6+x^5+2 x^4+10 x^3+2 x^2+x+33$
• $y^2=6 x^6+40 x^5+x^4+38 x^3+38 x^2+11 x+24$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.am $\times$ 1.43.ak 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.am : \(\Q(\sqrt{-7}) \).
• 1.43.ak : \(\Q(\sqrt{-2}) \).

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_abi	$2$	(not in LMFDB)
2.43.c_abi	$2$	(not in LMFDB)
2.43.w_hy	$2$	(not in LMFDB)