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

• Label: 2.83.a_gj
• 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:	$( 1 - x + 83 x^{2} )( 1 + x + 83 x^{2} )$
	$1 + 165 x^{2} + 6889 x^{4}$
Frobenius angles:	$\pm0.482521693698$, $\pm0.517478306302$
Angle rank:	$1$ (numerical)
Jacobians:	$24$
Isomorphism classes:	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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7055$	$49773025$	$326941455440$	$2251016163765625$	$15516041193925825775$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$7220$	$571788$	$47431428$	$3939040644$	$326942537510$	$27136050989628$	$2252292060328708$	$186940255267540404$	$15516041200645798100$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

• $y^2=53 x^6+24 x^5+82 x^4+43 x^3+19 x^2+32 x+13$
• $y^2=23 x^6+48 x^5+81 x^4+3 x^3+38 x^2+64 x+26$
• $y^2=8 x^6+2 x^5+48 x^4+40 x^3+7 x^2+20 x+67$
• $y^2=16 x^6+4 x^5+13 x^4+80 x^3+14 x^2+40 x+51$
• $y^2=24 x^6+35 x^5+20 x^4+62 x^3+66 x^2+45 x+71$
• $y^2=48 x^6+70 x^5+40 x^4+41 x^3+49 x^2+7 x+59$
• $y^2=3 x^6+5 x^5+65 x^4+54 x^3+77 x^2+19 x+37$
• $y^2=6 x^6+10 x^5+47 x^4+25 x^3+71 x^2+38 x+74$
• $y^2=82 x^6+27 x^5+82 x^4+8 x^3+50 x^2+21 x+2$
• $y^2=81 x^6+54 x^5+81 x^4+16 x^3+17 x^2+42 x+4$
• $y^2=24 x^6+3 x^5+69 x^4+38 x^3+69 x^2+3 x+24$
• $y^2=48 x^6+6 x^5+55 x^4+76 x^3+55 x^2+6 x+48$
• $y^2=79 x^6+54 x^5+51 x^4+65 x^3+38 x^2+34 x+76$
• $y^2=75 x^6+25 x^5+19 x^4+47 x^3+76 x^2+68 x+69$
• $y^2=32 x^6+11 x^5+9 x^4+13 x^3+75 x^2+63 x+71$
• $y^2=64 x^6+22 x^5+18 x^4+26 x^3+67 x^2+43 x+59$
• $y^2=76 x^6+2 x^5+59 x^4+56 x^3+4 x^2+60 x+35$
• $y^2=69 x^6+4 x^5+35 x^4+29 x^3+8 x^2+37 x+70$
• $y^2=21 x^6+46 x^5+47 x^4+65 x^3+18 x^2+53 x+70$
• $y^2=42 x^6+9 x^5+11 x^4+47 x^3+36 x^2+23 x+57$
• $y^2=73 x^6+29 x^5+3 x^4+50 x^3+10 x^2+64 x+60$
• $y^2=63 x^6+58 x^5+6 x^4+17 x^3+20 x^2+45 x+37$
• $y^2=43 x^6+73 x^5+29 x^4+56 x^3+70 x^2+35 x+46$
• $y^2=3 x^6+63 x^5+58 x^4+29 x^3+57 x^2+70 x+9$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{83}$

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

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

The base change of $A$ to $\F_{83^{2}}$ is 1.6889.gj^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-331}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.83.ac_gl	$2$	(not in LMFDB)
2.83.c_gl	$2$	(not in LMFDB)
2.83.a_agj	$4$	(not in LMFDB)