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

• Label: 2.43.av_hi
• 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:	$( 1 - 13 x + 43 x^{2} )( 1 - 8 x + 43 x^{2} )$
	$1 - 21 x + 190 x^{2} - 903 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.0421616081610$, $\pm0.291171725172$
Angle rank:	$1$ (numerical)
Jacobians:	$6$
Isomorphism classes:	30
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1116$	$3307824$	$6321251664$	$11688275491776$	$21609164073030516$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$23$	$1789$	$79508$	$3418825$	$146992673$	$6321140278$	$271816950155$	$11688193440529$	$502592611936844$	$21611482524392989$

Jacobians and polarizations

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

• $y^2=3 x^6+9 x^3+21$
• $y^2=5 x^6+33 x^5+5 x^4+30 x^3+22 x^2+13 x+26$
• $y^2=17 x^6+40 x^5+24 x^4+22 x^3+26 x^2+5 x+37$
• $y^2=2 x^6+11 x^5+11 x^4+11 x^2+36 x+2$
• $y^2=17 x^6+5 x^5+34 x^4+23 x^3+22 x^2+35 x+12$
• $y^2=9 x^6+32 x^5+28 x^4+36 x^3+9 x^2+35 x+26$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{43^{6}}$.

Endomorphism algebra over $\F_{43}$

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

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

The base change of $A$ to $\F_{43^{6}}$ is 1.6321363049.agiuc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{43^{2}}$
  The base change of $A$ to $\F_{43^{2}}$ is 1.1849.adf $\times$ 1.1849.w. The endomorphism algebra for each factor is:

  • 1.1849.adf : \(\Q(\sqrt{-3}) \).
  • 1.1849.w : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{43^{3}}$
  The base change of $A$ to $\F_{43^{3}}$ is 1.79507.aua $\times$ 1.79507.ua. The endomorphism algebra for each factor is:

  • 1.79507.aua : \(\Q(\sqrt{-3}) \).
  • 1.79507.ua : \(\Q(\sqrt{-3}) \).

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.af_as	$2$	(not in LMFDB)
2.43.f_as	$2$	(not in LMFDB)
2.43.as_fv	$3$	(not in LMFDB)
2.43.ad_bu	$3$	(not in LMFDB)
2.43.a_adf	$3$	(not in LMFDB)
2.43.a_w	$3$	(not in LMFDB)
2.43.a_cj	$3$	(not in LMFDB)
2.43.d_bu	$3$	(not in LMFDB)
2.43.s_fv	$3$	(not in LMFDB)
2.43.v_hi	$3$	(not in LMFDB)