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

• Label: 2.43.as_gg
• Base field: $\F_{43}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{43}$
Dimension:	$2$
L-polynomial:	$1 - 18 x + 162 x^{2} - 774 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.172484797803$, $\pm0.327515202197$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{5})\)
Galois group:	$C_2^2$
Jacobians:	$32$
Isomorphism classes:	43
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1220$	$3420880$	$6368767220$	$11702419974400$	$21613226197800500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$26$	$1850$	$80102$	$3422958$	$147020306$	$6321363050$	$271818807902$	$11688205316638$	$502592648826026$	$21611482313284250$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{43^{4}}$.

Endomorphism algebra over $\F_{43}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{5})\).

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

The base change of $A$ to $\F_{43^{4}}$ is 1.3418801.dby^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{43^{2}}$

  The base change of $A$ to $\F_{43^{2}}$ is the simple isogeny class 2.1849.a_dby and its endomorphism algebra is \(\Q(i, \sqrt{5})\).

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.s_gg	$2$	(not in LMFDB)
2.43.s_gg	$4$	(not in LMFDB)
2.43.a_acy	$8$	(not in LMFDB)