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

• Label: 2.89.au_ks
• 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:	$( 1 - 10 x + 89 x^{2} )^{2}$
	$1 - 20 x + 278 x^{2} - 1780 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.322192315511$, $\pm0.322192315511$
Angle rank:	$1$ (numerical)
Jacobians:	$80$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6400$	$64000000$	$499340089600$	$3937813504000000$	$31181149811270560000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$70$	$8078$	$708310$	$62761758$	$5583957350$	$496978533038$	$44231316403190$	$3936588866233918$	$350356405958621830$	$31181719947090216398$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 80 curves (of which all are hyperelliptic):

• $y^2=11 x^6+73 x^5+15 x^4+35 x^3+60 x^2+11 x+81$
• $y^2=76 x^6+73 x^4+73 x^2+76$
• $y^2=61 x^6+60 x^5+15 x^4+13 x^3+82 x^2+19 x+52$
• $y^2=75 x^6+30 x^5+35 x^4+51 x^3+75 x^2+24 x+75$
• $y^2=35 x^6+78 x^5+29 x^4+58 x^3+88 x^2+39 x+67$
• $y^2=33 x^6+65 x^5+52 x^4+14 x^3+52 x^2+65 x+33$
• $y^2=32 x^6+52 x^5+57 x^4+32 x^3+80 x^2+65 x+16$
• $y^2=81 x^6+78 x^5+85 x^4+75 x^3+8 x^2+45 x+64$
• $y^2=22 x^6+26 x^5+46 x^3+70 x+44$
• $y^2=61 x^6+56 x^5+67 x^4+41 x^3+67 x^2+56 x+61$
• $y^2=24 x^6+72 x^5+3 x^4+21 x^3+23 x^2+49 x+56$
• $y^2=25 x^6+24 x^5+54 x^3+75 x+69$
• $y^2=6 x^6+19 x^5+27 x^4+56 x^3+59 x^2+47 x+53$
• $y^2=29 x^6+86 x^5+87 x^4+77 x^3+32 x^2+33 x+31$
• $y^2=58 x^6+20 x^5+33 x^4+32 x^3+33 x^2+20 x+58$
• $y^2=24 x^6+43 x^5+53 x^4+28 x^3+8 x^2+51 x+61$
• $y^2=68 x^6+19 x^5+64 x^4+36 x^3+21 x^2+51 x+11$
• $y^2=55 x^6+64 x^5+80 x^4+6 x^3+80 x^2+64 x+55$
• $y^2=25 x^6+44 x^5+50 x^4+36 x^3+68 x^2+84 x+82$
• $y^2=77 x^6+36 x^5+73 x^4+42 x^3+73 x^2+36 x+77$
• and

• $y^2=15 x^6+53 x^5+68 x^4+43 x^3+73 x^2+79 x+6$
• $y^2=66 x^6+3 x^5+69 x^4+86 x^3+69 x^2+84 x+7$
• $y^2=40 x^6+17 x^5+8 x^4+32 x^3+63 x^2+19 x+27$
• $y^2=54 x^6+39 x^5+59 x^4+82 x^3+12 x^2+81 x+77$
• $y^2=3 x^6+10 x^5+18 x^4+18 x^3+29 x^2+42 x$
• $y^2=67 x^6+62 x^5+35 x^4+23 x^3+46 x^2+47 x+62$
• $y^2=74 x^6+65 x^5+60 x^4+23 x^3+19 x^2+86 x+30$
• $y^2=13 x^5+9 x^4+35 x^3+57 x^2+36 x+69$
• $y^2=25 x^6+43 x^5+65 x^4+70 x^3+80 x^2+70 x+59$
• $y^2=69 x^6+19 x^4+19 x^2+69$
• $y^2=24 x^6+37 x^5+13 x^4+48 x^3+13 x^2+37 x+24$
• $y^2=74 x^6+81 x^5+85 x^4+64 x^3+32 x^2+56 x+59$
• $y^2=68 x^6+19 x^5+72 x^4+10 x^3+32 x^2+51 x+78$
• $y^2=17 x^6+55 x^5+85 x^4+84 x^3+9 x^2+17 x+8$
• $y^2=59 x^6+62 x^5+32 x^4+83 x^3+47 x^2+28 x+14$
• $y^2=74 x^6+17 x^5+77 x^4+22 x^3+33 x^2+34 x+38$
• $y^2=81 x^6+64 x^4+64 x^2+81$
• $y^2=11 x^6+68 x^5+x^4+74 x^3+x^2+68 x+11$
• $y^2=7 x^6+82 x^5+31 x^4+43 x^3+35 x^2+66 x+3$
• $y^2=80 x^6+x^5+60 x^4+19 x^3+83 x^2+81 x+64$
• $y^2=11 x^6+72 x^5+8 x^4+43 x^3+8 x^2+72 x+11$
• $y^2=76 x^6+7 x^5+32 x^4+40 x^3+32 x^2+7 x+76$
• $y^2=74 x^6+7 x^5+51 x^4+29 x^3+5 x^2+83 x+62$
• $y^2=68 x^6+74 x^5+36 x^4+72 x^3+36 x^2+74 x+68$
• $y^2=41 x^6+66 x^5+13 x^4+19 x^3+56 x^2+24 x+58$
• $y^2=32 x^6+79 x^5+44 x^4+49 x^3+25 x^2+9 x+16$
• $y^2=81 x^6+77 x^5+85 x^4+71 x^3+81 x^2+43 x+79$
• $y^2=55 x^6+51 x^5+73 x^4+51 x^3+30 x^2+19 x+59$
• $y^2=51 x^6+50 x^5+12 x^4+80 x^3+52 x^2+85 x+3$
• $y^2=3 x^6+82 x^4+82 x^2+3$
• $y^2=56 x^6+51 x^5+39 x^4+35 x^3+39 x^2+51 x+56$
• $y^2=11 x^6+61 x^5+66 x^4+24 x^3+64 x^2+47 x+68$
• $y^2=14 x^6+43 x^5+71 x^4+18 x^3+58 x^2+19 x+18$
• $y^2=49 x^6+78 x^5+41 x^4+48 x^3+3 x^2+48 x+52$
• $y^2=57 x^6+21 x^5+5 x^4+18 x^3+5 x^2+21 x+57$
• $y^2=49 x^6+17 x^5+15 x^4+7 x^3+43 x^2+42 x+57$
• $y^2=65 x^5+7 x^4+5 x^3+7 x^2+65 x$
• $y^2=61 x^6+41 x^5+7 x^4+19 x^3+48 x^2+72 x+84$
• $y^2=29 x^6+56 x^4+7 x^3+56 x^2+29$
• $y^2=27 x^6+30 x^5+76 x^4+14 x^3+43 x^2+37 x+48$
• $y^2=28 x^6+40 x^5+55 x^4+12 x^3+55 x^2+40 x+28$
• $y^2=51 x^5+74 x^4+59 x^3+74 x^2+51 x$
• $y^2=31 x^6+47 x^4+47 x^2+31$
• $y^2=2 x^6+79 x^5+39 x^4+14 x^3+39 x^2+79 x+2$
• $y^2=63 x^6+49 x^4+49 x^2+63$
• $y^2=29 x^6+83 x^5+16 x^4+36 x^3+16 x^2+83 x+29$
• $y^2=27 x^6+64 x^5+45 x^4+54 x^3+44 x^2+20 x+50$
• $y^2=72 x^6+62 x^5+70 x^4+41 x^3+62 x^2+84 x+26$
• $y^2=59 x^6+66 x^5+24 x^4+72 x^3+41 x^2+86 x+62$
• $y^2=3 x^6+39 x^5+35 x^4+87 x^3+54 x^2+57 x+12$
• $y^2=82 x^6+18 x^5+15 x^4+77 x^3+19 x^2+64 x+5$
• $y^2=46 x^5+66 x^4+84 x^3+53 x^2+18 x+85$
• $y^2=70 x^6+43 x^5+27 x^4+8 x^3+63 x^2+66 x+52$
• $y^2=73 x^6+42 x^5+80 x^4+72 x^3+26 x^2+47 x+64$
• $y^2=51 x^6+43 x^5+54 x^4+61 x^3+22 x^2+12 x+45$
• $y^2=46 x^6+45 x^4+45 x^2+46$
• $y^2=35 x^6+85 x^5+43 x^4+79 x^3+59 x^2+2 x+6$
• $y^2=12 x^6+71 x^5+71 x^4+14 x^3+81 x^2+36 x+19$
• $y^2=69 x^6+28 x^5+80 x^4+87 x^3+60 x^2+13 x+52$
• $y^2=51 x^6+62 x^5+26 x^4+31 x^3+40 x^2+76 x+57$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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_da	$2$	(not in LMFDB)
2.89.u_ks	$2$	(not in LMFDB)
2.89.k_l	$3$	(not in LMFDB)