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

• Label: 2.83.abe_os
• Base field: $\F_{83}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{83}$
Dimension:	$2$
L-polynomial:	$( 1 - 18 x + 83 x^{2} )( 1 - 12 x + 83 x^{2} )$
	$1 - 30 x + 382 x^{2} - 2490 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.0496118990883$, $\pm0.271155063531$
Angle rank:	$2$ (numerical)
Jacobians:	$30$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4752$	$46531584$	$326888355024$	$2252392220491776$	$15515905884131066832$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$6754$	$571698$	$47460430$	$3939006294$	$326939250418$	$27136036291698$	$2252292120115294$	$186940254961923894$	$15516041191968600514$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

• $y^2=82 x^6+49 x^5+63 x^4+23 x^3+63 x^2+49 x+82$
• $y^2=82 x^6+11 x^5+20 x^4+55 x^3+25 x^2+22 x+34$
• $y^2=18 x^6+82 x^5+57 x^4+79 x^3+57 x^2+82 x+18$
• $y^2=46 x^6+29 x^5+33 x^4+79 x^3+40 x^2+75 x+5$
• $y^2=51 x^6+49 x^5+44 x^4+65 x^3+78 x^2+54 x+73$
• $y^2=27 x^6+20 x^5+9 x^4+48 x^3+79 x^2+33 x+46$
• $y^2=39 x^6+13 x^5+72 x^4+42 x^3+72 x^2+13 x+39$
• $y^2=69 x^6+21 x^5+69 x^4+36 x^3+58 x^2+24 x+74$
• $y^2=73 x^6+46 x^5+63 x^4+15 x^3+58 x^2+81 x+35$
• $y^2=70 x^6+19 x^5+5 x^4+16 x^3+39 x^2+67 x+12$
• $y^2=36 x^6+67 x^5+66 x^4+62 x^3+65 x^2+8 x+28$
• $y^2=36 x^6+24 x^5+18 x^4+12 x^3+48 x^2+36 x+11$
• $y^2=5 x^6+49 x^5+36 x^4+57 x^3+36 x^2+49 x+5$
• $y^2=73 x^6+38 x^5+51 x^4+20 x^3+53 x^2+11 x+82$
• $y^2=16 x^6+79 x^5+56 x^4+4 x^3+74 x^2+18 x+59$
• $y^2=24 x^6+54 x^5+62 x^4+57 x^3+59 x^2+55 x+57$
• $y^2=19 x^6+28 x^5+2 x^4+8 x^3+9 x^2+58 x+60$
• $y^2=13 x^6+82 x^5+61 x^4+40 x^3+65 x^2+35 x+5$
• $y^2=40 x^6+74 x^5+44 x^4+81 x^3+3 x^2+80 x+16$
• $y^2=60 x^6+x^5+5 x^4+65 x^3+38 x^2+18 x+13$
• $y^2=29 x^6+27 x^5+51 x^4+59 x^3+43 x^2+35 x+31$
• $y^2=28 x^6+39 x^5+6 x^4+20 x^3+10 x^2+8 x+67$
• $y^2=79 x^6+58 x^5+29 x^4+72 x^3+29 x^2+58 x+79$
• $y^2=55 x^6+54 x^5+39 x^4+82 x^3+75 x^2+76 x+34$
• $y^2=18 x^6+56 x^5+64 x^4+80 x^3+24 x^2+41 x+42$
• $y^2=39 x^6+59 x^5+29 x^4+46 x^3+36 x^2+18 x+74$
• $y^2=35 x^6+12 x^5+77 x^4+73 x^3+28 x^2+49 x+62$
• $y^2=79 x^6+37 x^5+4 x^4+45 x^3+76 x^2+74 x+22$
• $y^2=78 x^6+30 x^5+9 x^4+9 x^3+15 x^2+79 x+52$
• $y^2=67 x^6+10 x^5+79 x^4+44 x^3+28 x^2+51 x+24$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.as $\times$ 1.83.am and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.83.as : \(\Q(\sqrt{-2}) \).
• 1.83.am : \(\Q(\sqrt{-47}) \).

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_aby	$2$	(not in LMFDB)
2.83.g_aby	$2$	(not in LMFDB)
2.83.be_os	$2$	(not in LMFDB)