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

• Label: 2.89.s_jz
• 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:	$( 1 + 9 x + 89 x^{2} )^{2}$
	$1 + 18 x + 259 x^{2} + 1602 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.658275487260$, $\pm0.658275487260$
Angle rank:	$1$ (numerical)
Jacobians:	$65$
Cyclic group of points:	no
Non-cyclic primes:	$3, 11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9801$	$64304361$	$494625263616$	$3937396214255625$	$31182737242213990521$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$108$	$8116$	$701622$	$62755108$	$5584241628$	$496978506286$	$44231343743772$	$3936588973904068$	$350356401406173798$	$31181719935708009556$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 65 curves (of which all are hyperelliptic):

• $y^2=45 x^6+20 x^5+50 x^4+53 x^3+8 x^2+71 x+22$
• $y^2=72 x^6+84 x^5+53 x^4+79 x^3+19 x^2+84 x+60$
• $y^2=10 x^6+10 x^5+81 x^4+55 x^3+45 x^2+68 x+29$
• $y^2=81 x^6+37 x^5+60 x^4+44 x^3+71 x^2+87 x+46$
• $y^2=16 x^6+10 x^5+22 x^4+53 x^3+39 x^2+34 x+22$
• $y^2=42 x^6+8 x^5+39 x^4+16 x^3+45 x^2+13 x+19$
• $y^2=3 x^6+87 x^5+20 x^4+86 x^3+26 x^2+48 x+9$
• $y^2=36 x^6+87 x^5+43 x^4+54 x^3+85 x^2+11 x+82$
• $y^2=7 x^6+23 x^5+32 x^4+27 x^3+48 x^2+38 x+62$
• $y^2=70 x^6+30 x^5+49 x^4+75 x^3+71 x^2+10 x+29$
• $y^2=88 x^6+52 x^5+43 x^4+14 x^3+x^2+61 x+6$
• $y^2=57 x^6+29 x^5+81 x^4+15 x^3+25 x^2+76 x+81$
• $y^2=12 x^6+65 x^5+70 x^4+74 x^3+x^2+28 x+28$
• $y^2=2 x^6+53 x^5+55 x^4+86 x^3+10 x^2+40 x+85$
• $y^2=34 x^6+77 x^5+80 x^3+58 x^2+17 x+66$
• $y^2=21 x^6+43 x^5+84 x^4+9 x^3+23 x^2+41 x+41$
• $y^2=72 x^6+54 x^5+64 x^4+9 x^3+5 x^2+52 x+87$
• $y^2=16 x^6+74 x^5+75 x^4+30 x^3+82 x^2+8 x+26$
• $y^2=22 x^6+27 x^5+29 x^4+14 x^3+6 x^2+51 x+71$
• $y^2=63 x^6+15 x^5+85 x^4+75 x^3+10 x^2+27 x+28$
• and

• $y^2=15 x^6+33 x^5+6 x^4+15 x^3+73 x^2+27 x+26$
• $y^2=47 x^6+14 x^5+62 x^4+23 x^3+63 x^2+7 x+49$
• $y^2=59 x^6+55 x^5+38 x^4+52 x^3+61 x^2+20 x+14$
• $y^2=36 x^6+23 x^5+63 x^4+47 x^3+59 x^2+48 x+9$
• $y^2=19 x^6+39 x^5+10 x^4+58 x^3+36 x^2+64 x+52$
• $y^2=30 x^6+15 x^5+69 x^4+29 x^3+15 x^2+74 x+10$
• $y^2=79 x^6+51 x^5+11 x^4+47 x^3+6 x^2+78 x+55$
• $y^2=57 x^6+9 x^5+85 x^4+23 x^3+2 x^2+69 x+4$
• $y^2=15 x^6+72 x^5+8 x^4+21 x^3+42 x^2+71 x+86$
• $y^2=39 x^6+86 x^5+33 x^4+4 x^3+33 x^2+86 x+39$
• $y^2=15 x^6+70 x^5+x^4+17 x^3+65 x^2+39 x+24$
• $y^2=11 x^6+38 x^5+40 x^4+15 x^3+71 x^2+74 x+81$
• $y^2=60 x^6+29 x^5+76 x^4+7 x^3+43 x^2+42 x+78$
• $y^2=53 x^6+54 x^5+3 x^4+59 x^3+35 x^2+52 x+1$
• $y^2=20 x^6+17 x^5+58 x^4+32 x^3+26 x^2+34 x+21$
• $y^2=42 x^6+16 x^5+25 x^4+29 x^3+84 x^2+22 x+73$
• $y^2=48 x^6+62 x^5+40 x^4+22 x^3+20 x^2+60 x+6$
• $y^2=85 x^6+35 x^5+84 x^4+7 x^3+74 x^2+61 x+26$
• $y^2=27 x^6+2 x^5+6 x^4+15 x^3+49 x^2+86 x+4$
• $y^2=74 x^6+70 x^5+66 x^4+30 x^3+60 x^2+63 x+12$
• $y^2=62 x^6+36 x^5+37 x^4+14 x^3+35 x^2+72 x+74$
• $y^2=81 x^6+50 x^5+10 x^4+70 x^3+11 x^2+16 x+10$
• $y^2=29 x^6+21 x^5+12 x^4+52 x^2+66 x+66$
• $y^2=87 x^6+36 x^5+5 x^4+36 x^3+30 x^2+61 x+10$
• $y^2=5 x^6+10 x^5+80 x^4+62 x^3+61 x^2+57 x+56$
• $y^2=5 x^6+61 x^5+79 x^4+49 x^3+67 x^2+14 x+39$
• $y^2=11 x^6+13 x^5+59 x^4+62 x^3+23 x^2+38 x+68$
• $y^2=63 x^6+55 x^5+35 x^4+32 x^3+54 x^2+55 x+26$
• $y^2=26 x^6+71 x^5+26 x^4+50 x^3+13 x^2+30 x+61$
• $y^2=46 x^6+68 x^5+18 x^4+88 x^3+41 x^2+5 x+59$
• $y^2=84 x^6+68 x^5+37 x^4+x^3+30 x^2+10 x+82$
• $y^2=80 x^6+87 x^5+66 x^4+65 x^3+75 x^2+25 x+53$
• $y^2=71 x^6+13 x^5+71 x^4+75 x^3+16 x^2+74 x+50$
• $y^2=78 x^6+16 x^5+23 x^4+27 x^3+60 x^2+50 x+9$
• $y^2=87 x^6+25 x^5+44 x^4+33 x^3+64 x^2+61 x+68$
• $y^2=18 x^6+85 x^5+32 x^4+57 x^3+47 x^2+39 x+44$
• $y^2=35 x^6+84 x^5+10 x^4+17 x^3+45 x^2+10 x+41$
• $y^2=36 x^6+6 x^5+10 x^4+65 x^3+16 x^2+31 x+35$
• $y^2=85 x^6+x^5+22 x^4+6 x^3+11 x^2+67 x+44$
• $y^2=50 x^6+30 x^5+59 x^4+39 x^3+42 x^2+64 x+50$
• $y^2=40 x^6+68 x^5+87 x^4+71 x^3+x^2+17 x+84$
• $y^2=68 x^6+82 x^5+69 x^4+39 x^3+55 x^2+75 x+71$
• $y^2=77 x^6+86 x^5+79 x^4+74 x^3+20 x^2+77 x+7$
• $y^2=47 x^6+33 x^5+2 x^4+86 x^3+85 x^2+23 x+20$
• $y^2=39 x^6+55 x^5+52 x^4+76 x^3+62 x^2+72 x+1$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.j^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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.as_jz	$2$	(not in LMFDB)
2.89.a_dt	$2$	(not in LMFDB)
2.89.aj_ai	$3$	(not in LMFDB)