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

• Label: 2.89.e_ha
• 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:	$( 1 + 2 x + 89 x^{2} )^{2}$
	$1 + 4 x + 182 x^{2} + 356 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.533804287064$, $\pm0.533804287064$
Angle rank:	$1$ (numerical)
Jacobians:	$92$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8464$	$65545216$	$496241349136$	$3934777896484864$	$31182565172436657424$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$94$	$8270$	$703918$	$62713374$	$5584210814$	$496983557486$	$44231316891086$	$3936588639990334$	$350356405641304222$	$31181719940846891150$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 92 curves (of which all are hyperelliptic):

• $y^2=63 x^6+72 x^5+13 x^4+51 x^3+13 x^2+72 x+63$
• $y^2=4 x^6+4 x^4+4 x^2+4$
• $y^2=44 x^6+29 x^5+66 x^4+88 x^3+36 x^2+45 x+78$
• $y^2=8 x^6+71 x^5+52 x^4+56 x^3+30 x+67$
• $y^2=58 x^6+52 x^5+65 x^4+65 x^3+62 x^2+66 x+55$
• $y^2=33 x^6+45 x^5+70 x^4+7 x^3+53 x^2+86 x+27$
• $y^2=8 x^6+78 x^5+24 x^4+66 x^3+87 x^2+39 x+65$
• $y^2=68 x^6+51 x^5+64 x^3+x^2+62 x+44$
• $y^2=83 x^6+2 x^5+18 x^4+25 x^3+62 x^2+19 x+59$
• $y^2=21 x^6+88 x^5+22 x^4+40 x^3+16 x^2+64 x+52$
• $y^2=45 x^6+79 x^5+26 x^4+83 x^3+53 x^2+36 x+76$
• $y^2=12 x^6+30 x^5+52 x^4+66 x^3+77 x^2+22 x+60$
• $y^2=16 x^6+67 x^5+32 x^4+78 x^3+19 x^2+57 x+41$
• $y^2=63 x^6+65 x^4+65 x^2+63$
• $y^2=42 x^6+56 x^5+54 x^4+34 x^3+61 x^2+58$
• $y^2=37 x^6+12 x^4+68 x^3+22 x^2+70 x+69$
• $y^2=34 x^6+43 x^5+36 x^4+17 x^3+74 x^2+20 x+29$
• $y^2=12 x^6+7 x^5+74 x^4+4 x^3+74 x^2+66 x+49$
• $y^2=69 x^6+79 x^5+60 x^4+36 x^3+76 x^2+72 x+83$
• $y^2=4 x^6+25 x^5+28 x^4+59 x^3+38 x^2+2 x+34$
• and

• $y^2=23 x^6+52 x^5+73 x^4+71 x^3+41 x^2+74 x+61$
• $y^2=63 x^6+7 x^5+20 x^4+47 x^3+47 x^2+46 x+74$
• $y^2=79 x^6+72 x^5+87 x^4+62 x^3+59 x^2+25 x+21$
• $y^2=28 x^6+65 x^5+76 x^4+76 x^3+42 x^2+44 x+8$
• $y^2=9 x^6+42 x^5+65 x^4+20 x^3+55 x^2+60 x+4$
• $y^2=57 x^6+19 x^5+42 x^4+65 x^3+70 x^2+29 x+1$
• $y^2=77 x^6+62 x^5+73 x^4+32 x^3+10 x^2+80 x+39$
• $y^2=46 x^6+75 x^5+73 x^4+45 x^3+68 x^2+55 x+5$
• $y^2=52 x^6+26 x^5+31 x^4+48 x^3+16 x^2+26 x+63$
• $y^2=5 x^6+69 x^5+62 x^4+59 x^3+62 x^2+69 x+5$
• $y^2=51 x^6+33 x^5+62 x^4+58 x^3+72 x^2+3 x+21$
• $y^2=18 x^6+44 x^5+46 x^4+57 x^3+52 x^2+32 x+25$
• $y^2=53 x^6+27 x^5+15 x^4+49 x^3+49 x^2+70 x+29$
• $y^2=39 x^6+9 x^5+30 x^4+44 x^3+19 x^2+42 x+13$
• $y^2=47 x^6+87 x^5+57 x^4+76 x^3+53 x^2+29 x+32$
• $y^2=5 x^6+53 x^5+58 x^4+41 x^3+20 x^2+38 x+53$
• $y^2=68 x^6+50 x^5+8 x^4+27 x^3+18 x^2+42 x+41$
• $y^2=62 x^6+31 x^5+50 x^4+31 x^3+10 x^2+76 x+66$
• $y^2=18 x^6+55 x^5+8 x^4+34 x^3+64 x^2+16 x+2$
• $y^2=20 x^6+23 x^5+x^4+74 x^3+52 x^2+54 x+81$
• $y^2=9 x^6+31 x^5+58 x^4+43 x^3+87 x^2+22 x+71$
• $y^2=20 x^6+67 x^5+35 x^4+65 x^3+44 x^2+35 x+4$
• $y^2=28 x^6+23 x^5+70 x^4+46 x^3+28 x^2+82 x+26$
• $y^2=54 x^6+26 x^5+2 x^4+8 x^3+60 x^2+51 x+39$
• $y^2=59 x^6+27 x^5+54 x^4+74 x^3+35 x^2+61 x+7$
• $y^2=81 x^6+22 x^5+74 x^4+26 x^3+61 x^2+81 x+31$
• $y^2=32 x^6+82 x^5+38 x^4+23 x^3+82 x^2+19 x+15$
• $y^2=78 x^6+53 x^5+86 x^4+9 x^3+81 x^2+16 x+80$
• $y^2=76 x^6+19 x^5+44 x^4+2 x^3+49 x^2+26 x+43$
• $y^2=86 x^6+x^5+17 x^4+8 x^3+36 x^2+79 x+66$
• $y^2=56 x^6+61 x^5+22 x^4+67 x^3+5 x^2+25 x+3$
• $y^2=28 x^6+7 x^5+42 x^4+38 x^3+42 x^2+7 x+28$
• $y^2=27 x^6+24 x^5+87 x^4+47 x^3+51 x^2+28 x+26$
• $y^2=67 x^6+19 x^5+5 x^4+4 x^3+5 x^2+19 x+67$
• $y^2=x^6+30 x^5+85 x^4+28 x^3+42 x^2+42 x+9$
• $y^2=6 x^6+75 x^5+85 x^4+19 x^3+53 x^2+16 x+2$
• $y^2=64 x^6+83 x^5+87 x^4+51 x^2+25 x+48$
• $y^2=67 x^6+6 x^5+59 x^4+15 x^3+44 x^2+3 x+81$
• $y^2=83 x^6+64 x^5+39 x^4+6 x^3+x^2+36 x+57$
• $y^2=48 x^6+33 x^5+30 x^4+32 x^3+35 x^2+82 x+7$
• $y^2=9 x^6+53 x^5+76 x^4+32 x^3+69 x^2+14 x+31$
• $y^2=27 x^6+39 x^5+65 x^4+37 x^3+63 x^2+46 x+27$
• $y^2=x^6+70 x^5+50 x^4+27 x^3+25 x^2+62 x+78$
• $y^2=66 x^6+31 x^5+35 x^4+55 x^3+35 x^2+31 x+66$
• $y^2=45 x^6+37 x^5+14 x^4+44 x^3+70 x^2+33 x+81$
• $y^2=34 x^6+24 x^5+36 x^4+88 x^3+40 x^2+73 x+85$
• $y^2=17 x^6+54 x^5+86 x^4+59 x^3+86 x^2+54 x+25$
• $y^2=68 x^6+87 x^5+18 x^4+35 x^3+84 x^2+14 x+64$
• $y^2=24 x^6+6 x^5+5 x^4+7 x^3+44 x^2+41 x+37$
• $y^2=73 x^6+34 x^5+7 x^4+26 x^3+19 x^2+59 x+33$
• $y^2=67 x^6+29 x^4+29 x^2+67$
• $y^2=64 x^6+x^5+49 x^4+30 x^3+88 x^2+52 x+39$
• $y^2=78 x^5+73 x^4+55 x^3+29 x^2+20 x+44$
• $y^2=60 x^6+74 x^5+54 x^4+87 x^3+54 x^2+74 x+60$
• $y^2=50 x^6+46 x^5+52 x^4+58 x^3+19 x^2+39 x+52$
• $y^2=46 x^6+86 x^5+42 x^4+47 x^3+42 x^2+86 x+46$
• $y^2=70 x^6+13 x^5+42 x^4+39 x^3+5 x^2+58 x+6$
• $y^2=26 x^6+26 x^5+x^4+67 x^3+85 x^2+60 x+27$
• $y^2=87 x^6+75 x^5+47 x^4+21 x^3+35 x^2+26 x+36$
• $y^2=26 x^6+34 x^5+56 x^4+62 x^3+46 x^2+10 x+62$
• $y^2=38 x^6+12 x^4+12 x^2+38$
• $y^2=35 x^6+82 x^5+64 x^4+33 x^3+22 x^2+80 x+62$
• $y^2=48 x^6+88 x^5+49 x^4+44 x^3+13 x^2+42 x+4$
• $y^2=53 x^6+10 x^4+10 x^2+53$
• $y^2=57 x^6+88 x^5+7 x^4+87 x^3+56 x^2+2 x+78$
• $y^2=10 x^6+54 x^5+2 x^4+30 x^3+69 x^2+60 x+57$
• $y^2=39 x^6+6 x^5+83 x^4+61 x^3+83 x^2+6 x+39$
• $y^2=27 x^6+54 x^5+66 x^4+26 x^3+49 x^2+33 x+2$
• $y^2=36 x^6+65 x^5+56 x^4+51 x^3+68 x^2+21 x+31$
• $y^2=50 x^6+32 x^5+54 x^4+18 x^3+80 x^2+88 x+82$
• $y^2=56 x^6+57 x^5+19 x^4+28 x^3+78 x^2+45 x+71$
• $y^2=5 x^6+60 x^5+70 x^4+18 x^3+27 x^2+88 x+79$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.c^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-22}) \)$)$

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.ae_ha	$2$	(not in LMFDB)
2.89.a_gs	$2$	(not in LMFDB)
2.89.ac_adh	$3$	(not in LMFDB)