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

• Label: 2.43.ak_eh
• 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:	$( 1 - 5 x + 43 x^{2} )^{2}$
	$1 - 10 x + 111 x^{2} - 430 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.375494941494$, $\pm0.375494941494$
Angle rank:	$1$ (numerical)
Jacobians:	$12$
Cyclic group of points:	no
Non-cyclic primes:	$3, 13$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1521$	$3651921$	$6404480784$	$11688049850841$	$21604875082868961$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$1972$	$80548$	$3418756$	$146963494$	$6321140278$	$271819430098$	$11688213951748$	$502592645091004$	$21611481891066772$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.af^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.a_cj	$2$	(not in LMFDB)
2.43.k_eh	$2$	(not in LMFDB)
2.43.aq_fu	$3$	(not in LMFDB)
2.43.an_ew	$3$	(not in LMFDB)
2.43.f_as	$3$	(not in LMFDB)
2.43.i_v	$3$	(not in LMFDB)
2.43.ba_jv	$3$	(not in LMFDB)