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

• Label: 2.83.a_ay
• 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:	$1 - 24 x^{2} + 6889 x^{4}$
Frobenius angles:	$\pm0.226908719411$, $\pm0.773091280589$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-142}, \sqrt{190})\)
Galois group:	$C_2^2$
Jacobians:	$48$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6866$	$47141956$	$326940855554$	$2253545590882576$	$15516041181979059986$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6842$	$571788$	$47484726$	$3939040644$	$326941337738$	$27136050989628$	$2252292073386718$	$186940255267540404$	$15516041176752266522$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

• $y^2=72 x^6+6 x^5+19 x^4+58 x^3+6 x^2+46 x+70$
• $y^2=61 x^6+12 x^5+38 x^4+33 x^3+12 x^2+9 x+57$
• $y^2=37 x^6+20 x^5+17 x^4+51 x^3+33 x^2+62 x+30$
• $y^2=74 x^6+40 x^5+34 x^4+19 x^3+66 x^2+41 x+60$
• $y^2=59 x^6+63 x^5+63 x^4+33 x^3+33 x^2+66 x+61$
• $y^2=35 x^6+43 x^5+43 x^4+66 x^3+66 x^2+49 x+39$
• $y^2=74 x^6+5 x^5+26 x^4+79 x^3+13 x^2+18 x+76$
• $y^2=65 x^6+10 x^5+52 x^4+75 x^3+26 x^2+36 x+69$
• $y^2=39 x^6+24 x^5+22 x^4+78 x^3+20 x^2+63 x+3$
• $y^2=78 x^6+48 x^5+44 x^4+73 x^3+40 x^2+43 x+6$
• $y^2=34 x^6+66 x^5+61 x^4+19 x^3+40 x^2+17 x+74$
• $y^2=68 x^6+49 x^5+39 x^4+38 x^3+80 x^2+34 x+65$
• $y^2=49 x^6+46 x^5+27 x^4+29 x^3+34 x^2+9 x+57$
• $y^2=15 x^6+9 x^5+54 x^4+58 x^3+68 x^2+18 x+31$
• $y^2=6 x^6+32 x^5+69 x^4+52 x^3+40 x^2+42 x+25$
• $y^2=12 x^6+64 x^5+55 x^4+21 x^3+80 x^2+x+50$
• $y^2=10 x^6+20 x^5+31 x^3+31 x^2+7 x+58$
• $y^2=20 x^6+40 x^5+62 x^3+62 x^2+14 x+33$
• $y^2=71 x^6+18 x^5+61 x^4+20 x^3+6 x^2+71 x+52$
• $y^2=59 x^6+36 x^5+39 x^4+40 x^3+12 x^2+59 x+21$
• and

• $y^2=47 x^6+4 x^5+59 x^4+34 x^3+28 x^2+2 x+38$
• $y^2=11 x^6+8 x^5+35 x^4+68 x^3+56 x^2+4 x+76$
• $y^2=16 x^6+55 x^5+71 x^4+80 x^3+23 x^2+44 x+17$
• $y^2=32 x^6+27 x^5+59 x^4+77 x^3+46 x^2+5 x+34$
• $y^2=62 x^6+36 x^5+17 x^4+58 x^3+8 x^2+25 x+81$
• $y^2=41 x^6+72 x^5+34 x^4+33 x^3+16 x^2+50 x+79$
• $y^2=39 x^6+48 x^5+19 x^4+50 x^3+29 x^2+36 x+82$
• $y^2=78 x^6+13 x^5+38 x^4+17 x^3+58 x^2+72 x+81$
• $y^2=64 x^6+26 x^5+76 x^4+50 x^3+78 x^2+42 x+81$
• $y^2=45 x^6+52 x^5+69 x^4+17 x^3+73 x^2+x+79$
• $y^2=7 x^6+27 x^5+53 x^4+54 x^3+4 x^2+50 x+22$
• $y^2=14 x^6+54 x^5+23 x^4+25 x^3+8 x^2+17 x+44$
• $y^2=51 x^6+34 x^5+15 x^4+39 x^3+4 x^2+11 x+2$
• $y^2=19 x^6+68 x^5+30 x^4+78 x^3+8 x^2+22 x+4$
• $y^2=24 x^6+61 x^5+60 x^4+72 x^3+72 x^2+64 x+80$
• $y^2=48 x^6+39 x^5+37 x^4+61 x^3+61 x^2+45 x+77$
• $y^2=57 x^6+58 x^5+x^4+79 x^3+11 x^2+5 x+61$
• $y^2=31 x^6+33 x^5+2 x^4+75 x^3+22 x^2+10 x+39$
• $y^2=41 x^6+43 x^5+12 x^4+24 x^3+30 x^2+70 x+35$
• $y^2=82 x^6+3 x^5+24 x^4+48 x^3+60 x^2+57 x+70$
• $y^2=51 x^6+72 x^5+42 x^4+56 x^3+59 x^2+42 x+65$
• $y^2=19 x^6+61 x^5+x^4+29 x^3+35 x^2+x+47$
• $y^2=67 x^6+82 x^5+45 x^4+66 x^3+79 x^2+64 x+9$
• $y^2=51 x^6+81 x^5+7 x^4+49 x^3+75 x^2+45 x+18$
• $y^2=37 x^6+60 x^5+72 x^4+70 x^3+81 x^2+52 x+63$
• $y^2=74 x^6+37 x^5+61 x^4+57 x^3+79 x^2+21 x+43$
• $y^2=76 x^6+10 x^5+16 x^4+10 x^3+66 x^2+26 x+45$
• $y^2=69 x^6+20 x^5+32 x^4+20 x^3+49 x^2+52 x+7$

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 \(\Q(\sqrt{-142}, \sqrt{190})\).

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

The base change of $A$ to $\F_{83^{2}}$ is 1.6889.ay^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6745}) \)$)$

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.a_y	$4$	(not in LMFDB)