Abelian variety isogeny class 2.89.c_gx over $\F_{89}$

• Label: 2.89.c_gx
• Base field: $\F_{89}$
• 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:	$\F_{89}$
Dimension:	$2$
L-polynomial:	$( 1 + x + 89 x^{2} )^{2}$
	$1 + 2 x + 179 x^{2} + 178 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.516878298350$, $\pm0.516878298350$
Angle rank:	$1$ (numerical)
Jacobians:	$66$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8281$	$65593801$	$496607727616$	$3934645792830025$	$31182157297372129321$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$92$	$8276$	$704438$	$62711268$	$5584137772$	$496983969326$	$44231325246508$	$3936588576976708$	$350356404794973542$	$31181719949235253556$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 66 curves (of which all are hyperelliptic):

• $y^2=23 x^6+20 x^5+16 x^4+70 x^3+46 x^2+76 x+52$
• $y^2=66 x^6+13 x^5+22 x^4+17 x^3+4 x^2+86 x+49$
• $y^2=40 x^6+81 x^5+52 x^4+29 x^3+70 x^2+35 x+71$
• $y^2=17 x^6+17 x^5+9 x^4+34 x^3+77 x^2+46 x+64$
• $y^2=12 x^6+26 x^5+62 x^4+61 x^3+23 x^2+29 x+35$
• $y^2=79 x^6+57 x^5+41 x^4+4 x^3+33 x^2+26 x+30$
• $y^2=31 x^6+32 x^5+74 x^4+11 x^3+33 x^2+73 x+43$
• $y^2=31 x^6+53 x^5+74 x^4+82 x^3+18 x^2+20 x+72$
• $y^2=11 x^6+14 x^5+68 x^4+6 x^3+8 x^2+43 x+84$
• $y^2=71 x^6+5 x^5+36 x^4+24 x^3+54 x^2+85$
• $y^2=16 x^6+69 x^5+6 x^4+69 x^3+84 x^2+70 x+57$
• $y^2=22 x^6+38 x^5+41 x^4+3 x^3+76 x^2+52 x+71$
• $y^2=73 x^6+27 x^5+71 x^4+80 x^3+25 x^2+15 x+53$
• $y^2=65 x^6+72 x^5+42 x^4+3 x^3+80 x^2+36 x+28$
• $y^2=73 x^6+62 x^5+67 x^4+60 x^3+51 x^2+x+73$
• $y^2=79 x^6+6 x^5+52 x^4+68 x^3+81 x^2+31 x+25$
• $y^2=8 x^6+23 x^5+37 x^4+46 x^3+14 x^2+75 x+79$
• $y^2=31 x^6+80 x^5+10 x^4+24 x^3+85 x^2+18 x+28$
• $y^2=59 x^6+79 x^5+5 x^4+88 x^3+20 x^2+18 x+38$
• $y^2=83 x^6+80 x^5+37 x^4+38 x^3+33 x^2+18 x+54$
• and

• $y^2=51 x^6+33 x^5+43 x^4+67 x^3+59 x^2+28 x+24$
• $y^2=48 x^6+83 x^5+3 x^4+26 x^3+57 x^2+2 x+87$
• $y^2=6 x^6+60 x^5+6 x^4+40 x^3+35 x^2+54 x+76$
• $y^2=74 x^6+51 x^5+33 x^4+82 x^3+9 x+88$
• $y^2=70 x^6+33 x^5+26 x^4+31 x^3+3 x^2+63 x+44$
• $y^2=64 x^6+5 x^5+23 x^4+18 x^3+69 x^2+14 x+30$
• $y^2=22 x^6+4 x^5+37 x^4+29 x^3+76 x^2+12 x+47$
• $y^2=18 x^6+12 x^5+82 x^4+59 x^3+72 x^2+73 x+68$
• $y^2=79 x^6+49 x^5+74 x^4+61 x^3+59 x^2+18 x+9$
• $y^2=17 x^6+55 x^5+7 x^4+7 x^3+75 x^2+54 x+70$
• $y^2=75 x^6+55 x^5+74 x^4+54 x^3+28 x^2+8 x+46$
• $y^2=46 x^6+41 x^5+25 x^4+47 x^3+71 x^2+11 x+32$
• $y^2=42 x^6+37 x^5+11 x^4+45 x^3+80 x^2+13 x+45$
• $y^2=5 x^6+66 x^5+6 x^4+9 x^3+5 x^2+67 x+2$
• $y^2=88 x^6+75 x^5+10 x^4+38 x^3+78 x^2+38 x+7$
• $y^2=42 x^6+63 x^5+88 x^4+72 x^3+4 x^2+12 x+67$
• $y^2=79 x^6+4 x^5+31 x^4+3 x^3+14 x^2+46 x+32$
• $y^2=72 x^6+16 x^5+65 x^4+9 x^3+72 x^2+54 x+56$
• $y^2=72 x^6+67 x^5+23 x^4+10 x^3+6 x^2+15 x+67$
• $y^2=31 x^6+x^5+35 x^4+54 x^3+87 x^2+15 x+59$
• $y^2=19 x^6+68 x^5+59 x^4+47 x^3+51 x^2+85 x+52$
• $y^2=9 x^6+43 x^5+39 x^4+52 x^3+3 x^2+41 x+53$
• $y^2=43 x^6+23 x^5+39 x^4+6 x^3+39 x^2+23 x+43$
• $y^2=36 x^6+32 x^5+85 x^4+7 x^3+19 x^2+88 x+73$
• $y^2=71 x^6+71 x^5+76 x^4+29 x^3+59 x^2+36 x+44$
• $y^2=68 x^6+35 x^5+17 x^4+84 x^3+84 x^2+60 x+42$
• $y^2=13 x^6+33 x^5+14 x^4+x^3+43 x^2+82 x+63$
• $y^2=26 x^6+44 x^5+48 x^4+81 x^3+75 x^2+36 x+5$
• $y^2=86 x^6+42 x^5+60 x^4+19 x^3+51 x^2+21 x+41$
• $y^2=8 x^6+5 x^5+84 x^4+50 x^3+4 x^2+21 x+55$
• $y^2=67 x^6+38 x^5+53 x^4+39 x^3+12 x^2+33 x+35$
• $y^2=79 x^6+43 x^5+22 x^4+75 x^3+x^2+51 x+41$
• $y^2=21 x^6+57 x^5+29 x^4+16 x^3+57 x^2+38 x+9$
• $y^2=65 x^6+86 x^5+77 x^4+68 x^3+26 x^2+23 x+76$
• $y^2=86 x^6+3 x^5+11 x^4+75 x^3+42 x^2+42 x+43$
• $y^2=35 x^6+63 x^5+24 x^4+2 x^3+48 x^2+43 x+2$
• $y^2=25 x^6+86 x^5+39 x^4+31 x^3+40 x^2+23 x+5$
• $y^2=37 x^6+18 x^5+80 x^4+21 x^3+20 x^2+83 x+52$
• $y^2=23 x^6+63 x^5+79 x^4+7 x^3+76 x^2+85 x+87$
• $y^2=88 x^6+62 x^5+53 x^4+39 x^3+33 x^2+76 x+57$
• $y^2=73 x^6+62 x^5+87 x^4+8 x^3+36 x^2+63 x+40$
• $y^2=11 x^6+83 x^5+38 x^4+43 x^3+63 x^2+37 x+77$
• $y^2=78 x^6+52 x^5+54 x^4+53 x^3+84 x^2+71 x+40$
• $y^2=65 x^6+11 x^5+8 x^4+5 x^3+78 x^2+40 x+32$
• $y^2=13 x^6+88 x^5+63 x^4+87 x^3+46 x^2+39 x+7$
• $y^2=39 x^6+33 x^5+6 x^4+82 x^3+55 x^2+58 x+48$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{89}$.

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.b^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-355}) \)$)$

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
2.89.ac_gx	$2$	(not in LMFDB)
2.89.a_gv	$2$	(not in LMFDB)
2.89.ab_adk	$3$	(not in LMFDB)