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

• Label: 2.89.ay_mk
• 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:	$( 1 - 12 x + 89 x^{2} )^{2}$
	$1 - 24 x + 322 x^{2} - 2136 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.280588346245$, $\pm0.280588346245$
Angle rank:	$1$ (numerical)
Jacobians:	$63$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6084$	$63297936$	$499065950916$	$3938432011997184$	$31182221034306245124$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$66$	$7990$	$707922$	$62771614$	$5584149186$	$496979753686$	$44231308461714$	$3936588625313854$	$350356403895436098$	$31181719948276146550$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 63 curves (of which all are hyperelliptic):

• $y^2=77 x^6+31 x^5+69 x^4+53 x^3+50 x^2+38 x+54$
• $y^2=41 x^6+3 x^5+6 x^4+44 x^3+48 x^2+22 x+2$
• $y^2=85 x^6+80 x^5+38 x^4+2 x^3+76 x^2+53 x+57$
• $y^2=63 x^6+63 x^5+19 x^4+68 x^3+19 x^2+63 x+63$
• $y^2=58 x^6+77 x^5+60 x^4+68 x^3+29 x^2+39 x+76$
• $y^2=53 x^6+66 x^5+11 x^4+33 x^3+76 x^2+72 x+85$
• $y^2=59 x^6+79 x^5+62 x^4+73 x^3+73 x^2+30 x+62$
• $y^2=43 x^6+45 x^5+8 x^4+54 x^3+2 x^2+74 x+45$
• $y^2=65 x^6+30 x^5+42 x^4+78 x^3+22 x^2+51 x+44$
• $y^2=17 x^6+8 x^5+66 x^4+34 x^3+66 x^2+8 x+17$
• $y^2=56 x^6+52 x^5+64 x^4+87 x^3+64 x^2+52 x+56$
• $y^2=69 x^6+69 x^5+78 x^4+29 x^3+68 x^2+47 x+25$
• $y^2=13 x^6+5 x^5+28 x^4+49 x^3+48 x^2+71 x+52$
• $y^2=17 x^6+33 x^5+25 x^4+72 x^3+83 x^2+12 x+66$
• $y^2=75 x^6+22 x^5+28 x^4+79 x^3+42 x^2+83 x+17$
• $y^2=49 x^6+47 x^5+84 x^4+14 x^3+68 x^2+53 x+34$
• $y^2=54 x^6+26 x^5+62 x^4+50 x^3+15 x^2+48 x+60$
• $y^2=38 x^6+32 x^5+67 x^4+34 x^3+86 x^2+17 x+22$
• $y^2=39 x^6+53 x^5+35 x^4+79 x^3+64 x^2+7 x+58$
• $y^2=70 x^6+31 x^5+2 x^4+56 x^3+44 x^2+52 x+74$
• and

• $y^2=17 x^6+29 x^5+34 x^4+47 x^3+68 x^2+27 x+47$
• $y^2=74 x^6+37 x^5+67 x^4+85 x^3+56 x^2+2 x+82$
• $y^2=41 x^6+5 x^5+84 x^4+73 x^3+84 x^2+5 x+41$
• $y^2=31 x^6+84 x^5+51 x^4+82 x^3+40 x^2+75 x+30$
• $y^2=19 x^6+77 x^5+4 x^4+76 x^3+7 x^2+86 x+61$
• $y^2=35 x^6+11 x^4+11 x^2+35$
• $y^2=77 x^6+29 x^5+11 x^4+57 x^3+10 x^2+40 x+74$
• $y^2=39 x^6+x^5+10 x^4+43 x^3+83 x^2+81 x+19$
• $y^2=59 x^6+12 x^5+86 x^4+49 x^3+69 x^2+11 x+74$
• $y^2=67 x^6+62 x^5+78 x^4+23 x^3+78 x^2+62 x+67$
• $y^2=48 x^6+75 x^5+33 x^4+72 x^3+6 x^2+61 x+70$
• $y^2=72 x^6+36 x^5+74 x^4+53 x^3+74 x^2+36 x+72$
• $y^2=44 x^6+76 x^5+69 x^4+49 x^3+61 x^2+23 x+78$
• $y^2=41 x^6+45 x^5+82 x^4+13 x^3+4 x^2+53 x+19$
• $y^2=26 x^6+29 x^5+27 x^4+65 x^3+27 x^2+29 x+26$
• $y^2=22 x^6+64 x^5+38 x^4+77 x^3+63 x^2+56 x+70$
• $y^2=3 x^6+21 x^5+52 x^4+30 x^3+52 x^2+21 x+3$
• $y^2=31 x^5+47 x^4+19 x^3+73 x^2+38 x$
• $y^2=70 x^6+59 x^5+x^4+41 x^3+25 x^2+29 x+29$
• $y^2=52 x^6+86 x^5+50 x^4+61 x^3+50 x^2+86 x+52$
• $y^2=7 x^6+12 x^5+38 x^4+78 x^3+33 x^2+40 x+20$
• $y^2=7 x^6+39 x^5+60 x^4+5 x^3+60 x^2+39 x+7$
• $y^2=75 x^6+31 x^5+80 x^4+75 x^3+15 x^2+62 x+46$
• $y^2=3 x^6+34 x^5+72 x^4+61 x^3+68 x^2+10 x+75$
• $y^2=54 x^6+32 x^5+62 x^4+87 x^3+62 x^2+32 x+54$
• $y^2=78 x^6+83 x^5+51 x^4+80 x^3+65 x^2+23 x+36$
• $y^2=41 x^6+53 x^5+54 x^4+70 x^3+84 x+66$
• $y^2=15 x^6+86 x^5+55 x^4+46 x^3+16 x^2+56 x+48$
• $y^2=41 x^6+30 x^5+88 x^4+31 x^3+49 x^2+29 x+13$
• $y^2=15 x^6+76 x^5+65 x^4+61 x^3+41 x^2+37 x+31$
• $y^2=59 x^6+60 x^5+45 x^4+25 x^3+88 x^2+81 x+60$
• $y^2=34 x^6+16 x^4+16 x^2+34$
• $y^2=44 x^6+77 x^5+19 x^4+20 x^3+65 x^2+6 x+84$
• $y^2=40 x^6+6 x^5+50 x^4+22 x^3+67 x^2+52 x+36$
• $y^2=66 x^6+73 x^5+15 x^4+83 x^3+65 x^2+16 x+70$
• $y^2=53 x^6+35 x^5+18 x^4+7 x^3+53 x^2+78 x+60$
• $y^2=27 x^6+43 x^5+46 x^4+34 x^3+46 x^2+43 x+27$
• $y^2=6 x^6+20 x^5+86 x^4+52 x^3+88 x^2+12 x+30$
• $y^2=42 x^6+65 x^5+82 x^4+73 x^3+26 x^2+14 x+2$
• $y^2=76 x^6+77 x^5+70 x^4+78 x^3+70 x^2+77 x+76$
• $y^2=23 x^6+50 x^5+50 x^4+11 x^3+82 x^2+36 x+72$
• $y^2=40 x^6+82 x^5+70 x^4+32 x^3+8 x^2+7 x+66$
• $y^2=10 x^6+63 x^4+63 x^2+10$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.am^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-53}) \)$)$

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_bi	$2$	(not in LMFDB)
2.89.y_mk	$2$	(not in LMFDB)
2.89.m_cd	$3$	(not in LMFDB)