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

• Label: 2.43.at_gi
• 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:	$( 1 - 13 x + 43 x^{2} )( 1 - 6 x + 43 x^{2} )$
	$1 - 19 x + 164 x^{2} - 817 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.0421616081610$, $\pm0.348746511119$
Angle rank:	$2$ (numerical)
Jacobians:	$4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1178$	$3357300$	$6324253208$	$11681389620000$	$21606201062297798$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$25$	$1817$	$79546$	$3416809$	$146972515$	$6321099314$	$271817793937$	$11688202335121$	$502592636006158$	$21611482253187257$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

• $y^2=18 x^6+6 x^5+28 x^4+39 x^3+13 x^2+9 x+34$
• $y^2=24 x^6+39 x^5+5 x^4+x^3+14 x^2+11 x+11$
• $y^2=29 x^6+35 x^5+18 x^4+3 x^3+13 x^2+18 x+3$
• $y^2=37 x^6+40 x^5+17 x^4+14 x^3+2 x^2+25 x+13$

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.ag 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.ag : \(\Q(\sqrt{-34}) \).

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.ah_i	$2$	(not in LMFDB)
2.43.h_i	$2$	(not in LMFDB)
2.43.t_gi	$2$	(not in LMFDB)
2.43.ab_ce	$3$	(not in LMFDB)
2.43.c_bm	$3$	(not in LMFDB)