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

• Label: 2.89.as_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:	$\pm0.341724512740$, $\pm0.341724512740$
Angle rank:	$1$ (numerical)
Jacobians:	$65$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6561$	$64304361$	$499345742736$	$3937396214255625$	$31180702656649026321$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$8116$	$708318$	$62755108$	$5583877272$	$496978506286$	$44231326047288$	$3936588973904068$	$350356406008796622$	$31181719935708009556$

Jacobians and polarizations

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

• $y^2=15 x^6+66 x^5+76 x^4+77 x^3+62 x^2+83 x+37$
• $y^2=24 x^6+28 x^5+77 x^4+56 x^3+36 x^2+28 x+20$
• $y^2=30 x^6+30 x^5+65 x^4+76 x^3+46 x^2+26 x+87$
• $y^2=65 x^6+22 x^5+2 x^4+43 x^3+35 x^2+83 x+49$
• $y^2=35 x^6+33 x^5+37 x^4+77 x^3+13 x^2+41 x+37$
• $y^2=37 x^6+24 x^5+28 x^4+48 x^3+46 x^2+39 x+57$
• $y^2=x^6+29 x^5+66 x^4+88 x^3+68 x^2+16 x+3$
• $y^2=19 x^6+83 x^5+40 x^4+73 x^3+77 x^2+33 x+68$
• $y^2=32 x^6+67 x^5+70 x^4+9 x^3+16 x^2+72 x+80$
• $y^2=32 x^6+x^5+58 x^4+47 x^3+35 x^2+30 x+87$
• $y^2=86 x^6+67 x^5+40 x^4+42 x^3+3 x^2+5 x+18$
• $y^2=82 x^6+87 x^5+65 x^4+45 x^3+75 x^2+50 x+65$
• $y^2=36 x^6+17 x^5+32 x^4+44 x^3+3 x^2+84 x+84$
• $y^2=6 x^6+70 x^5+76 x^4+80 x^3+30 x^2+31 x+77$
• $y^2=41 x^6+85 x^5+86 x^3+49 x^2+65 x+22$
• $y^2=7 x^6+44 x^5+28 x^4+3 x^3+67 x^2+73 x+73$
• $y^2=24 x^6+18 x^5+51 x^4+3 x^3+61 x^2+47 x+29$
• $y^2=35 x^6+84 x^5+25 x^4+10 x^3+57 x^2+62 x+68$
• $y^2=66 x^6+81 x^5+87 x^4+42 x^3+18 x^2+64 x+35$
• $y^2=21 x^6+5 x^5+58 x^4+25 x^3+33 x^2+9 x+39$
• and

• $y^2=5 x^6+11 x^5+2 x^4+5 x^3+54 x^2+9 x+68$
• $y^2=52 x^6+42 x^5+8 x^4+69 x^3+11 x^2+21 x+58$
• $y^2=88 x^6+76 x^5+25 x^4+67 x^3+5 x^2+60 x+42$
• $y^2=19 x^6+69 x^5+11 x^4+52 x^3+88 x^2+55 x+27$
• $y^2=57 x^6+28 x^5+30 x^4+85 x^3+19 x^2+14 x+67$
• $y^2=x^6+45 x^5+29 x^4+87 x^3+45 x^2+44 x+30$
• $y^2=56 x^6+17 x^5+63 x^4+75 x^3+2 x^2+26 x+48$
• $y^2=82 x^6+27 x^5+77 x^4+69 x^3+6 x^2+29 x+12$
• $y^2=45 x^6+38 x^5+24 x^4+63 x^3+37 x^2+35 x+80$
• $y^2=28 x^6+80 x^5+10 x^4+12 x^3+10 x^2+80 x+28$
• $y^2=5 x^6+53 x^5+30 x^4+65 x^3+81 x^2+13 x+8$
• $y^2=63 x^6+72 x^5+43 x^4+5 x^3+83 x^2+84 x+27$
• $y^2=20 x^6+69 x^5+55 x^4+32 x^3+44 x^2+14 x+26$
• $y^2=70 x^6+73 x^5+9 x^4+88 x^3+16 x^2+67 x+3$
• $y^2=66 x^6+65 x^5+49 x^4+70 x^3+68 x^2+41 x+7$
• $y^2=14 x^6+35 x^5+38 x^4+69 x^3+28 x^2+37 x+54$
• $y^2=16 x^6+80 x^5+43 x^4+37 x^3+66 x^2+20 x+2$
• $y^2=77 x^6+16 x^5+74 x^4+21 x^3+44 x^2+5 x+78$
• $y^2=81 x^6+6 x^5+18 x^4+45 x^3+58 x^2+80 x+12$
• $y^2=44 x^6+32 x^5+20 x^4+x^3+2 x^2+11 x+36$
• $y^2=80 x^6+12 x^5+42 x^4+64 x^3+71 x^2+24 x+84$
• $y^2=65 x^6+61 x^5+30 x^4+32 x^3+33 x^2+48 x+30$
• $y^2=69 x^6+7 x^5+4 x^4+47 x^2+22 x+22$
• $y^2=83 x^6+19 x^5+15 x^4+19 x^3+x^2+5 x+30$
• $y^2=61 x^6+33 x^5+86 x^4+80 x^3+50 x^2+19 x+78$
• $y^2=61 x^6+50 x^5+56 x^4+46 x^3+52 x^2+64 x+13$
• $y^2=33 x^6+39 x^5+88 x^4+8 x^3+69 x^2+25 x+26$
• $y^2=21 x^6+48 x^5+71 x^4+70 x^3+18 x^2+48 x+68$
• $y^2=78 x^6+35 x^5+78 x^4+61 x^3+39 x^2+x+5$
• $y^2=45 x^6+82 x^5+6 x^4+59 x^3+73 x^2+61 x+79$
• $y^2=74 x^6+26 x^5+22 x^4+3 x^3+x^2+30 x+68$
• $y^2=62 x^6+83 x^5+20 x^4+17 x^3+47 x^2+75 x+70$
• $y^2=83 x^6+34 x^5+83 x^4+25 x^3+35 x^2+84 x+76$
• $y^2=56 x^6+48 x^5+69 x^4+81 x^3+2 x^2+61 x+27$
• $y^2=83 x^6+75 x^5+43 x^4+10 x^3+14 x^2+5 x+26$
• $y^2=6 x^6+58 x^5+70 x^4+19 x^3+75 x^2+13 x+74$
• $y^2=16 x^6+74 x^5+30 x^4+51 x^3+46 x^2+30 x+34$
• $y^2=19 x^6+18 x^5+30 x^4+17 x^3+48 x^2+4 x+16$
• $y^2=58 x^6+30 x^5+37 x^4+2 x^3+63 x^2+52 x+74$
• $y^2=76 x^6+10 x^5+79 x^4+13 x^3+14 x^2+51 x+76$
• $y^2=43 x^6+82 x^5+29 x^4+83 x^3+30 x^2+65 x+28$
• $y^2=82 x^6+57 x^5+23 x^4+13 x^3+48 x^2+25 x+83$
• $y^2=53 x^6+80 x^5+59 x^4+44 x^3+60 x^2+53 x+21$
• $y^2=52 x^6+10 x^5+6 x^4+80 x^3+77 x^2+69 x+60$
• $y^2=28 x^6+76 x^5+67 x^4+50 x^3+8 x^2+38 x+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.aj^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.a_dt	$2$	(not in LMFDB)
2.89.s_jz	$2$	(not in LMFDB)
2.89.j_ai	$3$	(not in LMFDB)