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

• Label: 2.89.ac_adh
• Base field: $\F_{89}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{89}$
Dimension:	$2$
L-polynomial:	$1 - 2 x - 85 x^{2} - 178 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.132862379603$, $\pm0.799529046269$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-22})\)
Galois group:	$C_2^2$
Jacobians:	$48$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7657$	$61386169$	$496241349136$	$3937494572821225$	$31181297317322594377$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$88$	$7748$	$703918$	$62756676$	$5583983768$	$496983557486$	$44231343897752$	$3936588888557956$	$350356405641304222$	$31181719924525829828$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

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

• $y^2=76 x^6+51 x^5+60 x^4+54 x^3+76 x^2+71 x+33$
• $y^2=63 x^6+69 x^5+38 x^4+8 x^3+42 x^2+63 x+35$
• $y^2=69 x^6+24 x^5+77 x^4+73 x^3+44 x^2+26 x+25$
• $y^2=49 x^6+74 x^5+88 x^4+11 x^3+52 x^2+21 x+70$
• $y^2=65 x^6+73 x^5+81 x^4+53 x^3+10 x^2+9 x+62$
• $y^2=77 x^6+14 x^5+38 x^4+71 x^3+53 x^2+36 x+80$
• $y^2=67 x^6+57 x^5+60 x^4+5 x^3+30 x^2+52 x+50$
• $y^2=59 x^6+14 x^5+41 x^4+33 x^3+2 x^2+85 x+42$
• $y^2=77 x^6+82 x^5+21 x^4+5 x^3+40 x^2+57 x+59$
• $y^2=66 x^6+42 x^5+37 x^4+48 x^3+3 x^2+4 x+29$
• $y^2=x^6+18 x^5+85 x^4+18 x^3+71 x^2+74 x+6$
• $y^2=64 x^6+81 x^5+14 x^4+68 x^3+22 x^2+22 x+15$
• $y^2=59 x^6+60 x^5+54 x^4+81 x^3+23 x^2+57 x+29$
• $y^2=16 x^6+15 x^5+34 x^4+69 x^3+30 x^2+69 x+53$
• $y^2=34 x^6+76 x^5+16 x^4+62 x^3+60 x^2+44 x+56$
• $y^2=3 x^6+87 x^5+5 x^4+44 x^3+67 x^2+77 x+76$
• $y^2=8 x^6+33 x^5+39 x^4+28 x^3+5 x^2+87 x+20$
• $y^2=79 x^6+84 x^5+70 x^4+55 x^3+10 x^2+84 x+66$
• $y^2=40 x^6+43 x^5+42 x^4+33 x^3+28 x^2+82 x+50$
• $y^2=68 x^6+88 x^5+63 x^4+44 x^3+21 x^2+29 x+83$
• $y^2=80 x^6+54 x^5+42 x^4+63 x^3+58 x^2+61 x+7$
• $y^2=29 x^6+13 x^5+15 x^4+70 x^3+65 x^2+67 x+87$
• $y^2=13 x^6+70 x^5+11 x^4+17 x^3+14 x^2+81 x+19$
• $y^2=22 x^6+55 x^5+57 x^4+65 x^3+62 x^2+79 x+75$
• $y^2=74 x^6+68 x^5+63 x^4+46 x^3+14 x^2+5 x+6$
• $y^2=27 x^6+63 x^5+2 x^4+62 x^3+34 x^2+58 x+82$
• $y^2=45 x^6+54 x^5+58 x^4+36 x^3+2 x^2+53 x+75$
• $y^2=14 x^6+35 x^5+14 x^4+79 x^3+72 x^2+40 x+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{89^{3}}$.

Endomorphism algebra over $\F_{89}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-22})\).

Endomorphism algebra over $\overline{\F}_{89}$

The base change of $A$ to $\F_{89^{3}}$ is 1.704969.aug^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.c_adh	$2$	(not in LMFDB)
2.89.e_ha	$3$	(not in LMFDB)
2.89.ae_ha	$6$	(not in LMFDB)