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

• Label: 2.83.e_gn
• 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:	$( 1 + x + 83 x^{2} )( 1 + 3 x + 83 x^{2} )$
	$1 + 4 x + 169 x^{2} + 332 x^{3} + 6889 x^{4}$
Frobenius angles:	$\pm0.517478306302$, $\pm0.552648295368$
Angle rank:	$2$ (numerical)
Jacobians:	$36$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7395$	$49716585$	$326388204720$	$2251138381763625$	$15516539108721044475$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$88$	$7212$	$570820$	$47434004$	$3939167048$	$326942080614$	$27136037540504$	$2252292122971876$	$186940256539508860$	$15516041193271135932$

Jacobians and polarizations

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

• $y^2=51 x^6+10 x^5+77 x^4+80 x^3+10 x^2+37 x+59$
• $y^2=52 x^6+65 x^5+25 x^4+66 x^3+33 x^2+11 x+82$
• $y^2=51 x^6+5 x^5+43 x^4+34 x^3+15 x^2+2 x+38$
• $y^2=29 x^6+8 x^5+79 x^4+11 x^3+46 x^2+62 x+61$
• $y^2=9 x^6+20 x^5+48 x^4+28 x^3+59 x^2+5 x+30$
• $y^2=17 x^6+33 x^5+17 x^4+64 x^2+26 x+63$
• $y^2=48 x^6+55 x^5+52 x^4+64 x^3+50 x^2+8 x+16$
• $y^2=38 x^6+44 x^5+80 x^4+59 x^3+52 x^2+41 x+10$
• $y^2=58 x^6+10 x^5+11 x^4+74 x^3+10 x^2+70 x+22$
• $y^2=56 x^6+40 x^5+52 x^4+46 x^3+15 x^2+63 x+43$
• $y^2=26 x^6+27 x^5+69 x^4+20 x^3+7 x^2+69 x+59$
• $y^2=45 x^6+30 x^5+66 x^4+22 x^3+66 x^2+30 x+45$
• $y^2=55 x^6+82 x^5+28 x^4+30 x^3+25 x^2+19 x+60$
• $y^2=41 x^6+53 x^5+34 x^4+78 x^3+13 x^2+15 x+30$
• $y^2=36 x^6+13 x^5+5 x^4+38 x^3+55 x^2+79 x+25$
• $y^2=81 x^6+81 x^5+60 x^4+67 x^3+13 x^2+48 x+27$
• $y^2=25 x^6+71 x^5+32 x^4+23 x^3+39 x^2+24 x+48$
• $y^2=28 x^6+53 x^5+38 x^4+4 x^3+44 x^2+66 x+44$
• $y^2=35 x^6+70 x^5+42 x^4+52 x^3+55 x^2+68 x+5$
• $y^2=7 x^6+21 x^5+55 x^4+77 x^3+45 x^2+12 x+41$
• and

• $y^2=57 x^6+47 x^5+48 x^4+37 x^3+70 x^2+22 x+42$
• $y^2=4 x^6+8 x^5+10 x^4+32 x^3+3 x^2+14 x+27$
• $y^2=19 x^6+58 x^5+4 x^4+9 x^3+11 x^2+34 x+80$
• $y^2=26 x^6+31 x^5+40 x^4+23 x^3+10 x^2+59 x+3$
• $y^2=4 x^6+39 x^5+76 x^4+33 x^3+79 x^2+50 x+61$
• $y^2=65 x^6+25 x^5+40 x^4+71 x^3+69 x^2+29 x+1$
• $y^2=61 x^6+10 x^5+43 x^4+63 x^3+82 x^2+69 x+68$
• $y^2=20 x^6+45 x^5+59 x^4+77 x^3+77 x^2+8 x+47$
• $y^2=33 x^6+30 x^5+33 x^4+18 x^3+48 x^2+23 x+70$
• $y^2=20 x^6+3 x^5+64 x^4+10 x^3+10 x^2+59 x+58$
• $y^2=69 x^6+78 x^5+51 x^4+9 x^3+x^2+68 x+48$
• $y^2=78 x^6+23 x^5+15 x^4+6 x^3+57 x^2+30 x+51$
• $y^2=29 x^6+53 x^5+10 x^4+23 x^3+30 x^2+62 x+36$
• $y^2=43 x^6+x^5+62 x^4+6 x^3+22 x^2+25 x+20$
• $y^2=21 x^6+51 x^5+55 x^4+20 x^3+57 x^2+30 x+31$
• $y^2=5 x^6+2 x^5+10 x^4+49 x^3+3 x^2+45 x+13$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

The isogeny class factors as 1.83.b $\times$ 1.83.d 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.b : \(\Q(\sqrt{-331}) \).
• 1.83.d : \(\Q(\sqrt{-323}) \).

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.ae_gn	$2$	(not in LMFDB)
2.83.ac_gh	$2$	(not in LMFDB)
2.83.c_gh	$2$	(not in LMFDB)