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

• Label: 2.43.ay_iv
• 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:	$( 1 - 13 x + 43 x^{2} )( 1 - 11 x + 43 x^{2} )$
	$1 - 24 x + 229 x^{2} - 1032 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.0421616081610$, $\pm0.183291501244$
Angle rank:	$2$ (numerical)
Jacobians:	$2$
Isomorphism classes:	4
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1023$	$3207105$	$6287128848$	$11685744524025$	$21612113707735023$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$1732$	$79076$	$3418084$	$147012740$	$6321402934$	$271818642332$	$11688197654596$	$502592575051388$	$21611481987179332$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=30 x^6+9 x^5+28 x^4+8 x^3+33 x^2+4 x+28$
• $y^2=35 x^6+16 x^5+39 x^4+41 x^3+12 x^2+15 x+1$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.an $\times$ 1.43.al 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.an : \(\Q(\sqrt{-3}) \).
• 1.43.al : \(\Q(\sqrt{-51}) \).

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.ac_acf	$2$	(not in LMFDB)
2.43.c_acf	$2$	(not in LMFDB)
2.43.y_iv	$2$	(not in LMFDB)
2.43.ag_bf	$3$	(not in LMFDB)
2.43.ad_ac	$3$	(not in LMFDB)