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

• Label: 2.43.au_hd
• 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:	$( 1 - 11 x + 43 x^{2} )( 1 - 9 x + 43 x^{2} )$
	$1 - 20 x + 185 x^{2} - 860 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.183291501244$, $\pm0.259258415261$
Angle rank:	$2$ (numerical)
Jacobians:	$12$
Isomorphism classes:	16
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1155$	$3366825$	$6362904240$	$11709228155625$	$21617054566459275$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$24$	$1820$	$80028$	$3424948$	$147046344$	$6321486710$	$271818396888$	$11688194346148$	$502592569208964$	$21611482143169100$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.al $\times$ 1.43.aj 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.al : \(\Q(\sqrt{-51}) \).
• 1.43.aj : \(\Q(\sqrt{-91}) \).

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_an	$2$	(not in LMFDB)
2.43.c_an	$2$	(not in LMFDB)
2.43.u_hd	$2$	(not in LMFDB)