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

• Label: 2.89.ah_abo
• 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:	$1 - 7 x - 40 x^{2} - 623 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.0456819038149$, $\pm0.712348570482$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-307})\)
Galois group:	$C_2^2$
Jacobians:	$57$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7252$	$61729024$	$494833461136$	$3936638874649600$	$31180930319283417652$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$83$	$7793$	$701918$	$62743041$	$5583918043$	$496979453486$	$44231341049587$	$3936588680856001$	$350356403055062222$	$31181719938794004353$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 57 curves (of which all are hyperelliptic):

• $y^2=58 x^6+36 x^5+46 x^4+78 x^3+62 x^2+2 x+79$
• $y^2=76 x^5+68 x^4+19 x^3+57 x^2+69 x+57$
• $y^2=34 x^6+52 x^5+47 x^4+24 x^3+41 x^2+29 x+86$
• $y^2=56 x^6+5 x^4+45 x^3+77 x^2+49 x+29$
• $y^2=29 x^6+59 x^5+60 x^4+4 x^3+17 x^2+9 x+29$
• $y^2=15 x^6+83 x^5+49 x^4+37 x^3+37 x^2+79 x+3$
• $y^2=42 x^6+27 x^5+45 x^4+16 x^3+74 x^2+22 x+3$
• $y^2=55 x^6+22 x^5+73 x^4+31 x^3+41 x^2+57 x+65$
• $y^2=83 x^6+78 x^5+30 x^4+51 x^3+67 x^2+32 x+12$
• $y^2=12 x^6+18 x^5+50 x^4+19 x^3+83 x^2+29 x+21$
• $y^2=33 x^6+88 x^5+40 x^4+19 x^3+18 x^2+44 x+60$
• $y^2=15 x^6+30 x^5+33 x^4+5 x^3+64 x^2+27 x+20$
• $y^2=43 x^6+70 x^5+30 x^4+75 x^3+40 x^2+40 x+44$
• $y^2=6 x^6+72 x^5+77 x^4+52 x^3+16 x^2+44 x+16$
• $y^2=87 x^6+62 x^5+72 x^4+43 x^3+28 x^2+62$
• $y^2=17 x^6+47 x^5+55 x^4+53 x^3+81 x^2+26 x+76$
• $y^2=56 x^6+33 x^5+21 x^4+65 x^3+28 x+50$
• $y^2=59 x^6+74 x^5+11 x^4+65 x^3+16 x^2+23 x+27$
• $y^2=79 x^6+68 x^5+53 x^4+20 x^3+34 x^2+53 x+62$
• $y^2=39 x^6+67 x^5+84 x^4+16 x^3+58 x^2+53 x+45$
• and

• $y^2=2 x^6+8 x^5+73 x^4+65 x^3+16 x^2+72 x+46$
• $y^2=43 x^6+85 x^5+68 x^4+25 x^3+10 x^2+28 x+36$
• $y^2=51 x^6+21 x^5+42 x^4+38 x^3+3 x^2+74 x+3$
• $y^2=23 x^6+18 x^5+69 x^4+32 x^3+17 x^2+21 x+25$
• $y^2=68 x^6+38 x^5+39 x^4+x^3+49 x^2+45 x+85$
• $y^2=5 x^6+33 x^5+46 x^4+34 x^3+6 x^2+55 x+35$
• $y^2=42 x^5+20 x^4+61 x^3+5 x^2+17 x$
• $y^2=49 x^6+23 x^5+76 x^4+40 x^3+42 x^2+43 x+76$
• $y^2=15 x^6+4 x^5+82 x^4+35 x^3+71 x^2+23 x+67$
• $y^2=16 x^6+57 x^5+45 x^4+47 x^3+86 x^2+27 x+1$
• $y^2=49 x^6+46 x^5+81 x^4+6 x^3+27 x^2+67 x+68$
• $y^2=20 x^6+86 x^5+3 x^4+62 x^3+20 x^2+59 x+19$
• $y^2=84 x^6+15 x^5+67 x^4+42 x^3+43 x^2+73 x+24$
• $y^2=78 x^6+x^5+45 x^4+17 x^3+59 x^2+17 x+4$
• $y^2=45 x^6+39 x^5+67 x^4+2 x^3+57 x^2+79 x+9$
• $y^2=63 x^6+30 x^5+9 x^4+66 x^3+37 x^2+5 x+44$
• $y^2=18 x^6+7 x^5+52 x^4+32 x^3+24 x^2+20 x+1$
• $y^2=23 x^6+5 x^5+61 x^4+84 x^3+60 x^2+45 x+45$
• $y^2=76 x^6+11 x^5+59 x^4+18 x^3+82 x^2+25 x+29$
• $y^2=8 x^6+15 x^5+19 x^4+25 x^3+12 x^2+60 x+27$
• $y^2=46 x^6+37 x^5+85 x^4+x^3+26 x^2+21 x+66$
• $y^2=76 x^6+23 x^5+88 x^4+29 x^3+45 x^2+13 x+44$
• $y^2=58 x^6+79 x^5+39 x^4+79 x^3+16 x^2+42 x+28$
• $y^2=45 x^6+36 x^5+6 x^4+44 x^3+46 x^2+6 x+31$
• $y^2=77 x^6+80 x^5+85 x^4+18 x^3+83 x^2+23 x+85$
• $y^2=38 x^6+68 x^5+8 x^4+34 x^3+30 x^2+43 x+76$
• $y^2=67 x^6+83 x^5+31 x^4+83 x^3+59 x^2+24 x+56$
• $y^2=60 x^6+82 x^5+x^4+86 x^3+64 x^2+72 x+59$
• $y^2=56 x^6+50 x^5+46 x^4+70 x^3+46 x^2+48$
• $y^2=79 x^6+47 x^5+10 x^4+71 x^3+5 x^2+84 x+17$
• $y^2=62 x^6+45 x^5+68 x^4+74 x^3+25 x^2+x+20$
• $y^2=46 x^6+6 x^5+67 x^4+61 x^3+84 x^2+33 x+61$
• $y^2=55 x^6+16 x^5+49 x^4+37 x^3+6 x^2+12 x+83$
• $y^2=72 x^6+44 x^5+57 x^4+26 x^3+76 x^2+50 x+24$
• $y^2=13 x^6+17 x^5+15 x^4+74 x^3+16 x^2+26 x+63$
• $y^2=4 x^6+73 x^5+58 x^4+7 x^3+55 x^2+25 x$
• $y^2=64 x^6+35 x^5+36 x^4+36 x^3+31 x+75$

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 \(\Q(\sqrt{-3}, \sqrt{-307})\).

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

The base change of $A$ to $\F_{89^{3}}$ is 1.704969.acgs^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-307}) \)$)$

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.h_abo	$2$	(not in LMFDB)
2.89.o_it	$3$	(not in LMFDB)
2.89.ao_it	$6$	(not in LMFDB)