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

• Label: 2.43.w_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:	$( 1 + 10 x + 43 x^{2} )( 1 + 12 x + 43 x^{2} )$
	$1 + 22 x + 206 x^{2} + 946 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.776024765496$, $\pm0.867802827160$
Angle rank:	$2$ (numerical)
Jacobians:	$6$
Isomorphism classes:	24

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3024$	$3290112$	$6312723984$	$11701322809344$	$21606440776606224$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$66$	$1778$	$79398$	$3422638$	$146974146$	$6321564578$	$271817817414$	$11688201577246$	$502592619617154$	$21611482254800018$

Jacobians and polarizations

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

• $y^2=9 x^6+19 x^5+40 x^4+5 x^3+13 x^2+8 x+13$
• $y^2=7 x^6+7 x^5+28 x^4+6 x^3+32 x^2+3 x+28$
• $y^2=28 x^6+24 x^5+2 x^4+15 x^3+2 x^2+24 x+28$
• $y^2=11 x^6+8 x^5+41 x^4+36 x^3+41 x^2+8 x+11$
• $y^2=11 x^6+29 x^5+15 x^4+32 x^3+15 x^2+29 x+11$
• $y^2=2 x^6+42 x^5+29 x^4+27 x^3+27 x^2+18 x+8$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

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

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.aw_hy	$2$	(not in LMFDB)
2.43.ac_abi	$2$	(not in LMFDB)
2.43.c_abi	$2$	(not in LMFDB)