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

• Label: 2.43.ai_v
• 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:	$( 1 - 13 x + 43 x^{2} )( 1 + 5 x + 43 x^{2} )$
	$1 - 8 x + 21 x^{2} - 344 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.0421616081610$, $\pm0.624505058506$
Angle rank:	$1$ (numerical)
Jacobians:	$43$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1519$	$3376737$	$6239104144$	$11677219158969$	$21611974949815999$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$1828$	$78468$	$3415588$	$147011796$	$6321140278$	$271817575884$	$11688203769796$	$502592578782684$	$21611482030503268$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 43 curves (of which all are hyperelliptic):

• $y^2=18 x^6+34 x^5+11 x^4+12 x^3+33 x^2+24 x+26$
• $y^2=36 x^6+19 x^5+18 x^4+20 x^3+29 x^2+12 x+28$
• $y^2=3 x^6+34$
• $y^2=3 x^6+3 x^3+29$
• $y^2=36 x^6+41 x^5+x^4+8 x^3+13 x^2+6 x+15$
• $y^2=3 x^6+3 x^3+2$
• $y^2=30 x^6+3 x^5+15 x^4+18 x^3+6 x^2+32 x+10$
• $y^2=6 x^6+10 x^5+23 x^4+24 x^3+42 x^2+14 x+12$
• $y^2=17 x^6+34 x^5+29 x^4+24 x^3+36 x^2+10 x+28$
• $y^2=15 x^6+14 x^5+17 x^4+20 x^3+22 x^2+15 x+8$
• $y^2=18 x^6+28 x^5+11 x^4+12 x^3+14 x^2+2 x+2$
• $y^2=5 x^6+24 x^5+33 x^4+23 x^3+2 x^2+13 x+12$
• $y^2=8 x^6+18 x^5+12 x^4+42 x^3+x^2+24 x+21$
• $y^2=42 x^6+36 x^5+11 x^4+27 x^3+10 x^2+21 x+37$
• $y^2=30 x^6+12 x^5+36 x^4+36 x^3+26 x^2+17 x+18$
• $y^2=5 x^6+16 x^5+23 x^4+19 x^3+32 x^2+3 x+5$
• $y^2=29 x^6+21 x^5+x^4+31 x^3+33 x^2+28 x+13$
• $y^2=33 x^6+41 x^5+31 x^4+3 x^3+18 x^2+27 x+12$
• $y^2=21 x^6+42 x^5+9 x^4+28 x^3+40 x^2+34 x+37$
• $y^2=18 x^6+5 x^5+35 x^4+42 x^3+27 x^2+7 x+1$
• and

• $y^2=5 x^6+35 x^5+28 x^4+14 x^3+31 x^2+13 x+19$
• $y^2=32 x^6+3 x^5+21 x^4+18 x^3+6 x^2+26 x+28$
• $y^2=3 x^6+3 x^3+28$
• $y^2=26 x^6+21 x^5+6 x^4+33 x^3+5 x^2+27 x+23$
• $y^2=40 x^6+12 x^5+22 x^4+39 x^3+20 x^2+17$
• $y^2=34 x^6+32 x^5+29 x^4+18 x^3+12 x^2+14 x+35$
• $y^2=42 x^6+21 x^5+16 x^4+36 x^3+29 x^2+8 x+32$
• $y^2=4 x^6+39 x^5+24 x^4+39 x^3+5 x^2+15 x+14$
• $y^2=3 x^6+3 x^3+27$
• $y^2=42 x^6+33 x^5+5 x^4+34 x^2+23 x+12$
• $y^2=42 x^6+22 x^5+25 x^4+25 x^3+32 x+16$
• $y^2=42 x^6+27 x^5+29 x^4+37 x^3+20 x^2+28 x+33$
• $y^2=26 x^6+41 x^5+28 x^4+22 x^3+38 x^2+4 x+34$
• $y^2=35 x^6+3 x^5+29 x^4+15 x^3+25 x^2+15 x+26$
• $y^2=21 x^6+12 x^5+13 x^4+29 x^3+4 x^2+33 x+28$
• $y^2=2 x^6+6 x^5+4 x^4+33 x^3+29 x^2+15 x+30$
• $y^2=28 x^6+40 x^5+37 x^4+27 x^3+10 x^2+21 x+16$
• $y^2=30 x^6+26 x^5+4 x^4+38 x^3+32 x^2+30 x+31$
• $y^2=28 x^6+28 x^5+38 x^4+22 x^3+16 x^2+3 x+39$
• $y^2=16 x^6+22 x^5+19 x^4+22 x^3+36 x^2+40 x+2$
• $y^2=10 x^6+32 x^5+31 x^4+9 x^3+37 x^2+9 x+39$
• $y^2=5 x^6+34 x^5+6 x^4+32 x^3+41 x^2+7 x+26$
• $y^2=8 x^6+18 x^5+23 x^4+11 x^3+7 x^2+17 x+21$

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.an $\times$ 1.43.f 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.f : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{43^{3}}$ is 1.79507.aua^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.as_fv	$2$	(not in LMFDB)
2.43.i_v	$2$	(not in LMFDB)
2.43.s_fv	$2$	(not in LMFDB)
2.43.aba_jv	$3$	(not in LMFDB)
2.43.af_as	$3$	(not in LMFDB)
2.43.k_eh	$3$	(not in LMFDB)
2.43.n_ew	$3$	(not in LMFDB)
2.43.q_fu	$3$	(not in LMFDB)