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

• Label: 2.43.a_abc
• 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:	$1 - 28 x^{2} + 1849 x^{4}$
Frobenius angles:	$\pm0.197219993436$, $\pm0.802780006564$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-58}, \sqrt{114})\)
Galois group:	$C_2^2$
Jacobians:	$8$
Isomorphism classes:	32
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1822$	$3319684$	$6321496414$	$11708140384656$	$21611482020387982$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$44$	$1794$	$79508$	$3424630$	$147008444$	$6321629778$	$271818611108$	$11688196970014$	$502592611936844$	$21611481727491714$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{43^{2}}$.

Endomorphism algebra over $\F_{43}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-58}, \sqrt{114})\).

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

The base change of $A$ to $\F_{43^{2}}$ is 1.1849.abc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1653}) \)$)$

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.a_bc	$4$	(not in LMFDB)