Abelian variety isogeny class 1.43.ai over $\F_{43}$

• Label: 1.43.ai
• Base field: $\F_{43}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{43}$
Dimension:	$1$
L-polynomial:	$1 - 8 x + 43 x^{2}$
Frobenius angles:	$\pm0.291171725172$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}) \)
Galois group:	$C_2$
Jacobians:	$6$
Isomorphism classes:	6

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:	$1$
Slopes:	$[0, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$1872$	$80028$	$3422016$	$147011796$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$1872$	$80028$	$3422016$	$147011796$	$6321251664$	$271817575884$	$11688196785408$	$502592628513924$	$21611482596065232$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which 0 are hyperelliptic):

• $y^2=x^3+2 x+2$
• $y^2=x^3+6 x+6$
• $y^2=x^3+14 x+28$
• $y^2=x^3+11 x+22$
• $y^2=x^3+42 x+42$
• $y^2=x^3+1$

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 \(\Q(\sqrt{-3}) \).

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
1.43.i	$2$	(not in LMFDB)
1.43.af	$3$	(not in LMFDB)
1.43.n	$3$	(not in LMFDB)