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

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

Invariants

Base field:	$\F_{43}$
Dimension:	$2$
L-polynomial:	$1 - 18 x + 156 x^{2} - 774 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.111629996936$, $\pm0.357330234349$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-80 -18 \sqrt{11}})\)
Galois group:	$D_{4}$
Jacobians:	$10$
Cyclic group of points:	yes

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1214$	$3396772$	$6342835574$	$11688876696784$	$21609336494318774$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$26$	$1838$	$79778$	$3418998$	$146993846$	$6321300878$	$271819421774$	$11688212907166$	$502592691675674$	$21611482520782718$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{43}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-80 -18 \sqrt{11}})\).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.43.s_ga	$2$	(not in LMFDB)