Abelian variety isogeny class 2.83.a_afy over $\F_{83}$

• Label: 2.83.a_afy
• Base field: $\F_{83}$
• 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:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$1 - 154 x^{2} + 6889 x^{4}$
Frobenius angles:	$\pm0.0608868240001$, $\pm0.939113176000$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:	$C_2^2$
Jacobians:	$32$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6736$	$45373696$	$326939903824$	$2251349144211456$	$15516041189848086736$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6582$	$571788$	$47438446$	$3939040644$	$326939434278$	$27136050989628$	$2252292224444638$	$186940255267540404$	$15516041192490320022$

Jacobians and polarizations

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

• $y^2=x^6+8 x^5+42 x^4+57 x^3+9 x^2+19 x+22$
• $y^2=45 x^6+51 x^5+4 x^4+8 x^3+80 x^2+65 x+29$
• $y^2=7 x^6+56 x^5+75 x^4+71 x^3+6 x^2+15 x+21$
• $y^2=45 x^6+16 x^5+34 x^4+75 x^3+67 x^2+79 x+4$
• $y^2=72 x^6+53 x^5+55 x^4+33 x^3+63 x^2+38 x+48$
• $y^2=47 x^6+37 x^5+39 x^4+77 x^3+77 x^2+77 x+1$
• $y^2=47 x^6+3 x^5+20 x^4+80 x^3+21 x^2+28 x+29$
• $y^2=13 x^6+45 x^5+59 x^4+23 x^3+82 x^2+47 x+38$
• $y^2=67 x^6+10 x^5+49 x^4+12 x^3+71 x^2+9 x+33$
• $y^2=56 x^6+32 x^5+37 x^4+79 x^3+69 x^2+21 x+68$
• $y^2=36 x^6+81 x^5+47 x^4+57 x^3+27 x^2+30 x+73$
• $y^2=14 x^6+44 x^5+4 x^4+39 x^3+66 x^2+27 x+38$
• $y^2=72 x^6+42 x^5+41 x^4+27 x^3+14 x^2+60 x+25$
• $y^2=50 x^6+76 x^5+27 x^4+42 x^3+29 x^2+43 x+41$
• $y^2=51 x^6+19 x^5+23 x^4+41 x^3+61 x^2+77 x+10$
• $y^2=8 x^6+62 x^5+38 x^4+18 x^3+53 x^2+46 x+70$
• $y^2=66 x^6+73 x^5+40 x^4+49 x^3+50 x^2+69 x+39$
• $y^2=30 x^6+54 x^5+63 x^4+71 x^3+74 x^2+32 x+27$
• $y^2=12 x^6+20 x^5+82 x^4+50 x^3+51 x^2+62 x+45$
• $y^2=43 x^6+81 x^5+55 x^4+31 x^3+30 x^2+68 x+25$
• and

• $y^2=25 x^6+46 x^4+40 x^3+x^2+35$
• $y^2=49 x^6+23 x^5+39 x^4+21 x^3+65 x^2+27 x+67$
• $y^2=21 x^6+35 x^5+65 x^4+40 x^3+43 x^2+20 x+68$
• $y^2=27 x^6+51 x^5+56 x^4+23 x^3+12 x^2+7 x+13$
• $y^2=77 x^6+66 x^5+73 x^4+21 x^3+64 x^2+54 x+45$
• $y^2=35 x^6+75 x^5+49 x^4+65 x^3+2 x^2+33 x+3$
• $y^2=21 x^6+71 x^5+15 x^4+6 x^3+9 x^2+82 x+55$
• $y^2=11 x^6+12 x^5+59 x^4+12 x^3+56 x^2+28 x+4$
• $y^2=71 x^6+59 x^5+18 x^4+36 x^3+51 x^2+x+36$
• $y^2=49 x^6+7 x^5+20 x^4+20 x^3+17 x^2+22 x+21$
• $y^2=20 x^6+75 x^5+14 x^4+80 x^3+81 x^2+10 x+51$
• $y^2=30 x^6+x^5+77 x^4+8 x^3+76 x^2+59 x+73$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{83^{2}}$.

Endomorphism algebra over $\F_{83}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\).

Endomorphism algebra over $\overline{\F}_{83}$

The base change of $A$ to $\F_{83^{2}}$ is 1.6889.afy^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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
2.83.ag_dr	$3$	(not in LMFDB)
2.83.g_dr	$3$	(not in LMFDB)
2.83.a_fy	$4$	(not in LMFDB)