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

• Label: 2.83.a_ads
• 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:	$1 - 96 x^{2} + 6889 x^{4}$
Frobenius angles:	$\pm0.151855574760$, $\pm0.848144425240$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-70}, \sqrt{262})\)
Galois group:	$C_2^2$
Jacobians:	$36$
Isomorphism classes:	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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6794$	$46158436$	$326941472666$	$2252725357597456$	$15516041186746863914$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$6698$	$571788$	$47467446$	$3939040644$	$326942571962$	$27136050989628$	$2252292380348638$	$186940255267540404$	$15516041186287874378$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

• $y^2=43 x^6+56 x^5+39 x^4+59 x^3+23 x^2+41 x+62$
• $y^2=3 x^6+29 x^5+78 x^4+35 x^3+46 x^2+82 x+41$
• $y^2=77 x^6+79 x^5+10 x^4+32 x^3+47 x^2+30 x+23$
• $y^2=71 x^6+75 x^5+20 x^4+64 x^3+11 x^2+60 x+46$
• $y^2=80 x^6+34 x^5+2 x^4+22 x^3+21 x^2+47 x+24$
• $y^2=77 x^6+68 x^5+4 x^4+44 x^3+42 x^2+11 x+48$
• $y^2=3 x^6+13 x^5+68 x^4+59 x^3+81 x^2+27 x+36$
• $y^2=6 x^6+26 x^5+53 x^4+35 x^3+79 x^2+54 x+72$
• $y^2=74 x^6+61 x^5+40 x^4+52 x^3+4 x^2+60 x+56$
• $y^2=65 x^6+39 x^5+80 x^4+21 x^3+8 x^2+37 x+29$
• $y^2=41 x^6+36 x^5+69 x^4+57 x^3+65 x^2+24 x+25$
• $y^2=82 x^6+72 x^5+55 x^4+31 x^3+47 x^2+48 x+50$
• $y^2=13 x^6+47 x^5+79 x^4+63 x^3+36 x^2+33 x+28$
• $y^2=26 x^6+11 x^5+75 x^4+43 x^3+72 x^2+66 x+56$
• $y^2=61 x^6+64 x^5+56 x^3+52 x^2+21 x+67$
• $y^2=39 x^6+45 x^5+29 x^3+21 x^2+42 x+51$
• $y^2=76 x^6+x^5+2 x^4+45 x^3+27 x^2+9 x+79$
• $y^2=69 x^6+2 x^5+4 x^4+7 x^3+54 x^2+18 x+75$
• $y^2=66 x^6+22 x^5+74 x^4+50 x^3+27 x^2+32 x+53$
• $y^2=49 x^6+44 x^5+65 x^4+17 x^3+54 x^2+64 x+23$
• and

• $y^2=8 x^6+35 x^5+72 x^4+41 x^3+10 x^2+3 x+74$
• $y^2=16 x^6+70 x^5+61 x^4+82 x^3+20 x^2+6 x+65$
• $y^2=42 x^6+52 x^5+47 x^4+63 x^3+45 x^2+56 x+7$
• $y^2=x^6+21 x^5+11 x^4+43 x^3+7 x^2+29 x+14$
• $y^2=48 x^6+4 x^5+30 x^4+32 x^3+51 x^2+49 x+66$
• $y^2=13 x^6+8 x^5+60 x^4+64 x^3+19 x^2+15 x+49$
• $y^2=35 x^6+44 x^5+37 x^4+80 x^3+64 x^2+4 x+50$
• $y^2=70 x^6+5 x^5+74 x^4+77 x^3+45 x^2+8 x+17$
• $y^2=37 x^6+82 x^5+72 x^4+77 x^3+40 x^2+78 x+11$
• $y^2=74 x^6+81 x^5+61 x^4+71 x^3+80 x^2+73 x+22$
• $y^2=16 x^6+41 x^5+69 x^4+36 x^3+66 x^2+64 x+46$
• $y^2=32 x^6+82 x^5+55 x^4+72 x^3+49 x^2+45 x+9$
• $y^2=2 x^6+65 x^5+33 x^4+73 x^3+22 x^2+37 x+70$
• $y^2=4 x^6+47 x^5+66 x^4+63 x^3+44 x^2+74 x+57$
• $y^2=34 x^6+53 x^5+3 x^4+23 x^3+47 x^2+39 x+79$
• $y^2=68 x^6+23 x^5+6 x^4+46 x^3+11 x^2+78 x+75$

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 \(\Q(\sqrt{-70}, \sqrt{262})\).

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

The base change of $A$ to $\F_{83^{2}}$ is 1.6889.ads^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-4585}) \)$)$

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_ds	$4$	(not in LMFDB)