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

• Label: 2.89.ai_hm
• 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:	$( 1 - 4 x + 89 x^{2} )^{2}$
	$1 - 8 x + 194 x^{2} - 712 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.432002453901$, $\pm0.432002453901$
Angle rank:	$1$ (numerical)
Jacobians:	$60$
Cyclic group of points:	no
Non-cyclic primes:	$2, 43$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7396$	$65351056$	$498399288676$	$3935283749785600$	$31180257336762252196$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$82$	$8246$	$706978$	$62721438$	$5583797522$	$496982094806$	$44231361422498$	$3936588840267838$	$350356401484848082$	$31181719917999282806$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 60 curves (of which all are hyperelliptic):

• $y^2=38 x^6+55 x^5+75 x^4+20 x^3+76 x^2+17 x+65$
• $y^2=11 x^5+66 x^4+59 x^2+32 x+84$
• $y^2=17 x^6+23 x^5+60 x^4+47 x^3+39 x^2+54 x+12$
• $y^2=55 x^6+83 x^5+54 x^4+50 x^3+59 x^2+86 x+34$
• $y^2=62 x^6+5 x^5+82 x^4+80 x^3+15 x^2+39 x+48$
• $y^2=67 x^6+71 x^5+46 x^4+55 x^3+14 x^2+57 x+30$
• $y^2=30 x^6+13 x^5+55 x^4+33 x^3+79 x^2+26 x+46$
• $y^2=73 x^6+84 x^5+24 x^4+22 x^3+22 x^2+7 x+68$
• $y^2=3 x^6+53 x^5+82 x^4+34 x^3+48 x^2+82 x+46$
• $y^2=31 x^6+60 x^5+59 x^4+24 x^3+51 x^2+31 x+37$
• $y^2=45 x^6+62 x^5+8 x^4+73 x^3+75 x^2+12 x+60$
• $y^2=88 x^6+65 x^5+40 x^4+14 x^3+40 x^2+65 x+88$
• $y^2=82 x^6+66 x^5+56 x^4+86 x^3+x^2+18 x+83$
• $y^2=5 x^6+33 x^5+6 x^4+59 x^3+69 x^2+77 x+56$
• $y^2=63 x^6+15 x^5+67 x^4+84 x^3+64 x^2+74 x+26$
• $y^2=43 x^6+24 x^5+88 x^4+83 x^3+5 x^2+6 x+20$
• $y^2=84 x^6+12 x^5+58 x^4+72 x^3+58 x^2+12 x+84$
• $y^2=19 x^6+47 x^5+60 x^4+73 x^3+60 x^2+47 x+19$
• $y^2=87 x^6+50 x^5+74 x^4+54 x^3+24 x^2+21 x+48$
• $y^2=16 x^6+12 x^5+28 x^4+67 x^3+75 x^2+69 x+54$
• and

• $y^2=74 x^6+82 x^5+5 x^4+29 x^3+65 x^2+28 x+64$
• $y^2=75 x^6+35 x^5+85 x^4+86 x^3+17 x^2+20 x+41$
• $y^2=x^5+55 x^4+9 x^3+27 x^2+26 x+12$
• $y^2=14 x^6+75 x^5+48 x^4+53 x^3+20 x^2+25 x+25$
• $y^2=45 x^6+51 x^5+21 x^4+12 x^3+32 x^2+67 x+69$
• $y^2=43 x^6+76 x^4+76 x^2+43$
• $y^2=81 x^6+87 x^5+13 x^4+65 x^3+30 x^2+13 x+2$
• $y^2=24 x^6+77 x^5+61 x^4+87 x^3+8 x^2+64 x+72$
• $y^2=35 x^6+76 x^5+61 x^4+31 x^3+43 x^2+58 x+31$
• $y^2=38 x^6+54 x^5+12 x^4+2 x^3+67 x^2+63 x+68$
• $y^2=73 x^6+82 x^5+7 x^4+58 x^3+46 x^2+76 x+74$
• $y^2=46 x^6+39 x^5+86 x^4+45 x^3+49 x^2+64 x+77$
• $y^2=36 x^6+26 x^5+58 x^4+26 x^3+10 x^2+28 x+82$
• $y^2=85 x^6+50 x^5+22 x^4+58 x^3+12 x^2+20 x+23$
• $y^2=20 x^6+82 x^5+70 x^4+6 x^3+16 x^2+72 x+52$
• $y^2=65 x^6+28 x^5+39 x^4+79 x^3+34 x+83$
• $y^2=84 x^6+16 x^5+73 x^4+57 x^3+18 x^2+78 x+40$
• $y^2=74 x^6+61 x^5+57 x^4+47 x^3+57 x^2+61 x+74$
• $y^2=55 x^6+60 x^5+18 x^4+34 x^3+18 x^2+60 x+55$
• $y^2=14 x^6+58 x^5+8 x^4+20 x^3+34 x^2+46 x+76$
• $y^2=17 x^6+28 x^4+28 x^2+17$
• $y^2=29 x^6+7 x^5+3 x^4+66 x^3+40 x^2+65 x+19$
• $y^2=84 x^6+18 x^5+39 x^4+41 x^3+88 x^2+69 x+10$
• $y^2=26 x^6+10 x^5+67 x^4+16 x^3+46 x^2+11 x+32$
• $y^2=6 x^6+33 x^5+69 x^4+44 x^3+69 x^2+33 x+6$
• $y^2=19 x^6+26 x^5+53 x^4+74 x^3+34 x^2+67 x+65$
• $y^2=55 x^6+82 x^5+23 x^4+6 x^3+87 x^2+24 x+49$
• $y^2=80 x^6+85 x^5+66 x^4+26 x^3+19 x^2+55 x+83$
• $y^2=50 x^6+54 x^5+7 x^4+11 x^3+59 x^2+35 x+38$
• $y^2=58 x^6+21 x^5+70 x^4+13 x^3+69 x^2+22 x+1$
• $y^2=26 x^6+78 x^5+36 x^4+57 x^3+29 x^2+86 x+31$
• $y^2=56 x^6+11 x^5+76 x^4+51 x^3+74 x^2+88 x+83$
• $y^2=33 x^6+32 x^5+43 x^4+22 x^3+88 x^2+71 x+59$
• $y^2=22 x^6+78 x^5+37 x^4+38 x^3+64 x^2+21 x+49$
• $y^2=85 x^6+64 x^5+65 x^4+63 x^3+65 x^2+64 x+85$
• $y^2=45 x^6+14 x^5+46 x^4+61 x^3+13 x^2+62 x+18$
• $y^2=83 x^5+67 x^4+5 x^3+19 x^2+65 x+82$
• $y^2=52 x^6+59 x^5+42 x^4+42 x^3+42 x^2+59 x+52$
• $y^2=83 x^6+17 x^5+46 x^4+82 x^3+14 x^2+26 x+74$
• $y^2=84 x^6+39 x^5+7 x^4+32 x^3+83 x^2+45 x+24$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-85}) \)$)$

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_gg	$2$	(not in LMFDB)
2.89.i_hm	$2$	(not in LMFDB)
2.89.e_acv	$3$	(not in LMFDB)