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

• Label: 2.43.f_as
• 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:	$( 1 - 8 x + 43 x^{2} )( 1 + 13 x + 43 x^{2} )$
	$1 + 5 x - 18 x^{2} + 215 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.291171725172$, $\pm0.957838391839$
Angle rank:	$1$ (numerical)
Jacobians:	$13$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2052$	$3307824$	$6404480784$	$11688275491776$	$21614786686180332$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$49$	$1789$	$80548$	$3418825$	$147030919$	$6321140278$	$271818201613$	$11688193440529$	$502592645091004$	$21611482524392989$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 13 curves (of which all are hyperelliptic):

• $y^2=19 x^6+34 x^5+23 x^4+29 x^3+8 x^2+41 x+13$
• $y^2=27 x^6+33 x^5+41 x^4+37 x^3+2 x^2+19 x+11$
• $y^2=x^6+3 x^3+15$
• $y^2=34 x^6+18 x^5+31 x^4+38 x^3+32 x^2+39 x+21$
• $y^2=37 x^6+7 x^5+6 x^4+12 x^3+13 x^2+32 x+16$
• $y^2=26 x^6+41 x^5+15 x^4+22 x^3+37 x+26$
• $y^2=15 x^6+32 x^5+6 x^4+29 x^3+2 x^2+8 x+37$
• $y^2=25 x^6+16 x^5+11 x^4+41 x^3+16 x^2+25 x+17$
• $y^2=22 x^6+7 x^5+23 x^4+22 x^3+40 x^2+34 x+39$
• $y^2=38 x^6+20 x^5+22 x^4+17 x^3+38 x^2+29 x+18$
• $y^2=21 x^6+12 x^5+36 x^4+30 x^3+30 x^2+17 x+35$
• $y^2=25 x^6+36 x^5+10 x^4+20 x^3+32 x^2+13 x+9$
• $y^2=35 x^6+9 x^5+7 x^4+20 x^3+28 x^2+22 x+41$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{43^{3}}$.

Endomorphism algebra over $\F_{43}$

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

Endomorphism algebra over $\overline{\F}_{43}$

The base change of $A$ to $\F_{43^{3}}$ is 1.79507.ua^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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
2.43.av_hi	$2$	(not in LMFDB)
2.43.af_as	$2$	(not in LMFDB)
2.43.v_hi	$2$	(not in LMFDB)
2.43.aq_fu	$3$	(not in LMFDB)
2.43.an_ew	$3$	(not in LMFDB)
2.43.ak_eh	$3$	(not in LMFDB)
2.43.i_v	$3$	(not in LMFDB)
2.43.ba_jv	$3$	(not in LMFDB)