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

• Label: 2.89.s_jj
• 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:	$( 1 + 5 x + 89 x^{2} )( 1 + 13 x + 89 x^{2} )$
	$1 + 18 x + 243 x^{2} + 1602 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.585371785029$, $\pm0.741949407251$
Angle rank:	$2$ (numerical)
Jacobians:	$72$
Isomorphism classes:	204

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9785$	$64042825$	$495233096960$	$3937102921854025$	$31182021567541116425$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$108$	$8084$	$702486$	$62750436$	$5584113468$	$496981023662$	$44231333129532$	$3936588751002436$	$350356405121554374$	$31181719922751364724$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 72 curves (of which all are hyperelliptic):

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

• $y^2=36 x^6+77 x^5+33 x^4+7 x^3+36 x^2+2 x+4$
• $y^2=40 x^6+75 x^5+73 x^4+65 x^3+68 x^2+40 x+43$
• $y^2=28 x^6+43 x^5+23 x^4+53 x^3+78 x^2+54 x+55$
• $y^2=17 x^6+54 x^5+6 x^4+9 x^3+87 x^2+21 x+54$
• $y^2=22 x^6+57 x^5+47 x^4+49 x^3+81 x^2+39 x+68$
• $y^2=34 x^6+18 x^5+82 x^4+61 x^3+58 x^2+28 x+2$
• $y^2=78 x^6+6 x^5+77 x^4+14 x^3+15 x^2+38 x+55$
• $y^2=12 x^6+45 x^5+45 x^4+28 x^3+76 x^2+33 x+20$
• $y^2=9 x^6+3 x^5+34 x^4+53 x^3+65 x^2+5 x+76$
• $y^2=68 x^6+82 x^5+13 x^4+32 x^3+22 x^2+36 x+12$
• $y^2=79 x^6+32 x^5+33 x^4+32 x^3+26 x^2+28 x+77$
• $y^2=44 x^6+88 x^5+23 x^4+54 x^3+85 x^2+x+85$
• $y^2=3 x^6+57 x^5+74 x^4+79 x^3+20 x^2+46 x+25$
• $y^2=39 x^6+81 x^5+67 x^4+67 x^3+26 x^2+44 x+2$
• $y^2=17 x^6+57 x^5+30 x^4+68 x^3+15 x^2+81 x+80$
• $y^2=14 x^6+81 x^5+37 x^4+13 x^3+3 x^2+45 x+76$
• $y^2=x^6+80 x^5+87 x^4+76 x^3+21 x^2+9 x+55$
• $y^2=48 x^6+6 x^5+51 x^4+28 x^3+6 x^2+82 x+51$
• $y^2=4 x^6+58 x^5+77 x^4+29 x^3+74 x^2+7 x+13$
• $y^2=36 x^6+4 x^5+38 x^4+13 x^3+35 x^2+67 x+5$
• $y^2=50 x^6+21 x^5+26 x^4+70 x^3+16 x^2+76 x+45$
• $y^2=53 x^6+38 x^5+67 x^4+80 x^3+78 x^2+54 x+40$
• $y^2=23 x^6+16 x^5+34 x^4+40 x^3+74 x^2+69 x+40$
• $y^2=36 x^6+27 x^5+62 x^4+87 x^3+88 x^2+24 x+24$
• $y^2=22 x^6+29 x^5+60 x^4+78 x^3+52 x^2+67 x+68$
• $y^2=50 x^6+67 x^5+58 x^4+43 x^3+3 x^2+22 x+35$
• $y^2=44 x^6+37 x^5+82 x^4+56 x^3+46 x+49$
• $y^2=7 x^6+62 x^5+45 x^4+38 x^3+4 x^2+52 x+24$
• $y^2=68 x^6+32 x^5+86 x^4+35 x^3+55 x^2+31 x+3$
• $y^2=87 x^6+64 x^5+28 x^4+50 x^3+70 x^2+44 x+80$
• $y^2=x^6+2 x^5+22 x^4+82 x^3+5 x^2+88 x+10$
• $y^2=2 x^6+53 x^5+39 x^4+24 x^3+79 x^2+31 x+41$
• $y^2=14 x^6+49 x^5+57 x^4+4 x^2+22 x+71$
• $y^2=86 x^6+67 x^5+73 x^4+87 x^3+28 x^2+50 x+9$
• $y^2=86 x^6+29 x^5+19 x^4+4 x^3+54 x^2+12 x+49$
• $y^2=40 x^6+66 x^5+10 x^4+66 x^3+18 x^2+75 x+9$
• $y^2=5 x^6+67 x^5+6 x^4+62 x^3+75 x^2+78 x+9$
• $y^2=74 x^6+30 x^5+36 x^4+30 x^3+61 x^2+78 x+17$
• $y^2=32 x^6+58 x^5+62 x^4+2 x^3+48 x^2+x+17$
• $y^2=9 x^6+2 x^5+10 x^4+22 x^3+59 x^2+78 x+55$
• $y^2=3 x^6+19 x^5+7 x^4+84 x^3+19 x^2+31 x+26$
• $y^2=74 x^6+36 x^5+46 x^4+52 x^3+87 x^2+61 x+21$
• $y^2=56 x^6+8 x^5+78 x^4+88 x^3+14 x^2+52 x+55$
• $y^2=18 x^6+36 x^5+21 x^4+20 x^3+20 x^2+24 x+26$
• $y^2=80 x^6+85 x^5+79 x^4+44 x^3+56 x^2+61 x+4$
• $y^2=75 x^6+12 x^5+31 x^4+81 x^3+22 x^2+80 x+5$
• $y^2=25 x^6+33 x^5+41 x^4+78 x^3+58 x^2+6 x+12$
• $y^2=78 x^6+39 x^5+67 x^4+80 x^3+40 x^2+27 x+28$
• $y^2=26 x^6+11 x^5+87 x^4+34 x^3+68 x^2+48 x+14$
• $y^2=22 x^6+3 x^5+60 x^4+76 x^3+33 x^2+84 x+74$
• $y^2=2 x^6+41 x^5+56 x^4+9 x^3+87 x^2+16 x+50$
• $y^2=6 x^6+39 x^5+57 x^4+14 x^3+57 x^2+39 x+6$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.f $\times$ 1.89.n and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.89.f : \(\Q(\sqrt{-331}) \).
• 1.89.n : \(\Q(\sqrt{-187}) \).

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
2.89.as_jj	$2$	(not in LMFDB)
2.89.ai_ej	$2$	(not in LMFDB)
2.89.i_ej	$2$	(not in LMFDB)