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

• Label: 2.43.ay_iw
• 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:	$( 1 - 12 x + 43 x^{2} )^{2}$
	$1 - 24 x + 230 x^{2} - 1032 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.132197172840$, $\pm0.132197172840$
Angle rank:	$1$ (numerical)
Jacobians:	$3$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1024$	$3211264$	$6292931584$	$11690490986496$	$21614936855716864$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$1734$	$79148$	$3419470$	$147031940$	$6321616278$	$271820639516$	$11688213729694$	$502592686140404$	$21611482625287014$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=34 x^6+26 x^4+26 x^2+34$
• $y^2=21 x^6+11 x^5+13 x^4+8 x^3+13 x^2+11 x+21$
• $y^2=4 x^6+31 x^4+31 x^2+4$

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^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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.a_acg	$2$	(not in LMFDB)
2.43.y_iw	$2$	(not in LMFDB)
2.43.m_dx	$3$	(not in LMFDB)