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

• Label: 2.89.ac_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:	$\pm0.483121701650$, $\pm0.483121701650$
Angle rank:	$1$ (numerical)
Jacobians:	$66$
Cyclic group of points:	no
Non-cyclic primes:	$89$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7921$	$65593801$	$497357815696$	$3934645792830025$	$31181282587963643521$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$88$	$8276$	$705502$	$62711268$	$5583981128$	$496983969326$	$44231344544552$	$3936588576976708$	$350356402619996878$	$31181719949235253556$

Jacobians and polarizations

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

• $y^2=67 x^6+66 x^5+35 x^4+53 x^3+45 x^2+55 x+47$
• $y^2=20 x^6+39 x^5+66 x^4+51 x^3+12 x^2+80 x+58$
• $y^2=43 x^6+27 x^5+47 x^4+69 x^3+53 x^2+71 x+83$
• $y^2=65 x^6+65 x^5+3 x^4+41 x^3+85 x^2+45 x+51$
• $y^2=36 x^6+78 x^5+8 x^4+5 x^3+69 x^2+87 x+16$
• $y^2=59 x^6+82 x^5+34 x^4+12 x^3+10 x^2+78 x+1$
• $y^2=4 x^6+7 x^5+44 x^4+33 x^3+10 x^2+41 x+40$
• $y^2=4 x^6+70 x^5+44 x^4+68 x^3+54 x^2+60 x+38$
• $y^2=63 x^6+64 x^5+82 x^4+2 x^3+62 x^2+44 x+28$
• $y^2=35 x^6+15 x^5+19 x^4+72 x^3+73 x^2+77$
• $y^2=35 x^6+23 x^5+2 x^4+23 x^3+28 x^2+53 x+19$
• $y^2=66 x^6+25 x^5+34 x^4+9 x^3+50 x^2+67 x+35$
• $y^2=41 x^6+81 x^5+35 x^4+62 x^3+75 x^2+45 x+70$
• $y^2=17 x^6+38 x^5+37 x^4+9 x^3+62 x^2+19 x+84$
• $y^2=54 x^6+80 x^5+52 x^4+20 x^3+17 x^2+30 x+54$
• $y^2=59 x^6+18 x^5+67 x^4+26 x^3+65 x^2+4 x+75$
• $y^2=62 x^6+67 x^5+42 x^4+45 x^3+64 x^2+25 x+56$
• $y^2=4 x^6+62 x^5+30 x^4+72 x^3+77 x^2+54 x+84$
• $y^2=79 x^6+56 x^5+61 x^4+59 x^3+66 x^2+6 x+72$
• $y^2=87 x^6+86 x^5+42 x^4+72 x^3+11 x^2+6 x+18$
• and

• $y^2=64 x^6+10 x^5+40 x^4+23 x^3+88 x^2+84 x+72$
• $y^2=16 x^6+87 x^5+x^4+68 x^3+19 x^2+60 x+29$
• $y^2=18 x^6+2 x^5+18 x^4+31 x^3+16 x^2+73 x+50$
• $y^2=84 x^6+17 x^5+11 x^4+57 x^3+3 x+59$
• $y^2=32 x^6+10 x^5+78 x^4+4 x^3+9 x^2+11 x+43$
• $y^2=51 x^6+61 x^5+67 x^4+6 x^3+23 x^2+64 x+10$
• $y^2=37 x^6+31 x^5+42 x^4+69 x^3+55 x^2+4 x+75$
• $y^2=54 x^6+36 x^5+68 x^4+88 x^3+38 x^2+41 x+26$
• $y^2=59 x^6+58 x^5+44 x^4+5 x^3+88 x^2+54 x+27$
• $y^2=51 x^6+76 x^5+21 x^4+21 x^3+47 x^2+73 x+32$
• $y^2=47 x^6+76 x^5+44 x^4+73 x^3+84 x^2+24 x+49$
• $y^2=45 x^6+73 x^5+38 x^4+75 x^3+83 x^2+63 x+70$
• $y^2=37 x^6+22 x^5+33 x^4+46 x^3+62 x^2+39 x+46$
• $y^2=15 x^6+20 x^5+18 x^4+27 x^3+15 x^2+23 x+6$
• $y^2=86 x^6+47 x^5+30 x^4+25 x^3+56 x^2+25 x+21$
• $y^2=37 x^6+11 x^5+86 x^4+38 x^3+12 x^2+36 x+23$
• $y^2=56 x^6+31 x^5+40 x^4+x^3+64 x^2+45 x+70$
• $y^2=24 x^6+35 x^5+81 x^4+3 x^3+24 x^2+18 x+78$
• $y^2=24 x^6+52 x^5+67 x^4+33 x^3+2 x^2+5 x+52$
• $y^2=40 x^6+30 x^5+71 x^4+18 x^3+29 x^2+5 x+79$
• $y^2=57 x^6+26 x^5+88 x^4+52 x^3+64 x^2+77 x+67$
• $y^2=27 x^6+40 x^5+28 x^4+67 x^3+9 x^2+34 x+70$
• $y^2=44 x^6+67 x^5+13 x^4+2 x^3+13 x^2+67 x+44$
• $y^2=12 x^6+70 x^5+58 x^4+32 x^3+36 x^2+59 x+54$
• $y^2=35 x^6+35 x^5+50 x^4+87 x^3+88 x^2+19 x+43$
• $y^2=26 x^6+16 x^5+51 x^4+74 x^3+74 x^2+2 x+37$
• $y^2=39 x^6+10 x^5+42 x^4+3 x^3+40 x^2+68 x+11$
• $y^2=78 x^6+43 x^5+55 x^4+65 x^3+47 x^2+19 x+15$
• $y^2=88 x^6+14 x^5+20 x^4+36 x^3+17 x^2+7 x+73$
• $y^2=62 x^6+61 x^5+28 x^4+76 x^3+31 x^2+7 x+48$
• $y^2=52 x^6+72 x^5+77 x^4+13 x^3+4 x^2+11 x+71$
• $y^2=56 x^6+44 x^5+37 x^4+25 x^3+30 x^2+17 x+73$
• $y^2=7 x^6+19 x^5+69 x^4+35 x^3+19 x^2+72 x+3$
• $y^2=17 x^6+80 x^5+53 x^4+26 x^3+78 x^2+69 x+50$
• $y^2=80 x^6+9 x^5+33 x^4+47 x^3+37 x^2+37 x+40$
• $y^2=71 x^6+21 x^5+8 x^4+60 x^3+16 x^2+44 x+60$
• $y^2=75 x^6+80 x^5+28 x^4+4 x^3+31 x^2+69 x+15$
• $y^2=22 x^6+54 x^5+62 x^4+63 x^3+60 x^2+71 x+67$
• $y^2=67 x^6+21 x^5+56 x^4+32 x^3+55 x^2+58 x+29$
• $y^2=59 x^6+80 x^5+77 x^4+13 x^3+11 x^2+55 x+19$
• $y^2=54 x^6+80 x^5+29 x^4+62 x^3+12 x^2+21 x+43$
• $y^2=63 x^6+87 x^5+72 x^4+44 x^3+21 x^2+42 x+85$
• $y^2=56 x^6+67 x^5+73 x^4+70 x^3+74 x^2+35 x+31$
• $y^2=81 x^6+63 x^5+62 x^4+61 x^3+26 x^2+43 x+70$
• $y^2=39 x^6+86 x^5+11 x^4+83 x^3+49 x^2+28 x+21$
• $y^2=28 x^6+10 x^5+18 x^4+68 x^3+76 x^2+85 x+55$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.ab^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.a_gv	$2$	(not in LMFDB)
2.89.c_gx	$2$	(not in LMFDB)
2.89.b_adk	$3$	(not in LMFDB)