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

• Label: 2.89.y_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:	$\pm0.719411653755$, $\pm0.719411653755$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10404$	$63297936$	$494903808036$	$3938432011997184$	$31181218851988634724$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$114$	$7990$	$702018$	$62771614$	$5583969714$	$496979753686$	$44231361329346$	$3936588625313854$	$350356403519534322$	$31181719948276146550$

Jacobians and polarizations

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

• $y^2=53 x^6+4 x^5+29 x^4+70 x^3+61 x^2+25 x+73$
• $y^2=73 x^6+x^5+2 x^4+74 x^3+16 x^2+37 x+60$
• $y^2=77 x^6+62 x^5+25 x^4+6 x^3+50 x^2+70 x+82$
• $y^2=11 x^6+11 x^5+57 x^4+26 x^3+57 x^2+11 x+11$
• $y^2=85 x^6+53 x^5+2 x^4+26 x^3+87 x^2+28 x+50$
• $y^2=77 x^6+22 x^5+63 x^4+11 x^3+55 x^2+24 x+58$
• $y^2=79 x^6+56 x^5+80 x^4+54 x^3+54 x^2+10 x+80$
• $y^2=40 x^6+46 x^5+24 x^4+73 x^3+6 x^2+44 x+46$
• $y^2=81 x^6+10 x^5+14 x^4+26 x^3+37 x^2+17 x+74$
• $y^2=65 x^6+62 x^5+22 x^4+41 x^3+22 x^2+62 x+65$
• $y^2=78 x^6+47 x^5+51 x^4+29 x^3+51 x^2+47 x+78$
• $y^2=29 x^6+29 x^5+56 x^4+87 x^3+26 x^2+52 x+75$
• $y^2=34 x^6+61 x^5+39 x^4+46 x^3+16 x^2+83 x+47$
• $y^2=51 x^6+10 x^5+75 x^4+38 x^3+71 x^2+36 x+20$
• $y^2=47 x^6+66 x^5+84 x^4+59 x^3+37 x^2+71 x+51$
• $y^2=46 x^6+75 x^5+28 x^4+64 x^3+82 x^2+77 x+41$
• $y^2=73 x^6+78 x^5+8 x^4+61 x^3+45 x^2+55 x+2$
• $y^2=25 x^6+7 x^5+23 x^4+13 x^3+80 x^2+51 x+66$
• $y^2=13 x^6+77 x^5+71 x^4+56 x^3+51 x^2+32 x+49$
• $y^2=32 x^6+4 x^5+6 x^4+79 x^3+43 x^2+67 x+44$
• and

• $y^2=65 x^6+69 x^5+41 x^4+75 x^3+82 x^2+9 x+75$
• $y^2=44 x^6+22 x^5+23 x^4+77 x^3+79 x^2+6 x+68$
• $y^2=73 x^6+61 x^5+28 x^4+54 x^3+28 x^2+61 x+73$
• $y^2=40 x^6+28 x^5+17 x^4+57 x^3+43 x^2+25 x+10$
• $y^2=36 x^6+85 x^5+31 x^4+55 x^3+32 x^2+88 x+50$
• $y^2=16 x^6+33 x^4+33 x^2+16$
• $y^2=85 x^6+69 x^5+63 x^4+19 x^3+33 x^2+43 x+84$
• $y^2=28 x^6+3 x^5+30 x^4+40 x^3+71 x^2+65 x+57$
• $y^2=88 x^6+36 x^5+80 x^4+58 x^3+29 x^2+33 x+44$
• $y^2=23 x^6+8 x^5+56 x^4+69 x^3+56 x^2+8 x+23$
• $y^2=16 x^6+25 x^5+11 x^4+24 x^3+2 x^2+50 x+53$
• $y^2=24 x^6+12 x^5+84 x^4+77 x^3+84 x^2+12 x+24$
• $y^2=74 x^6+55 x^5+23 x^4+46 x^3+50 x^2+67 x+26$
• $y^2=73 x^6+15 x^5+57 x^4+34 x^3+31 x^2+77 x+36$
• $y^2=68 x^6+69 x^5+9 x^4+81 x^3+9 x^2+69 x+68$
• $y^2=37 x^6+51 x^5+72 x^4+85 x^3+21 x^2+78 x+53$
• $y^2=9 x^6+63 x^5+67 x^4+x^3+67 x^2+63 x+9$
• $y^2=40 x^5+75 x^4+36 x^3+54 x^2+72 x$
• $y^2=53 x^6+79 x^5+30 x^4+73 x^3+38 x^2+69 x+69$
• $y^2=67 x^6+80 x^5+61 x^4+5 x^3+61 x^2+80 x+67$
• $y^2=32 x^6+4 x^5+72 x^4+26 x^3+11 x^2+43 x+66$
• $y^2=21 x^6+28 x^5+2 x^4+15 x^3+2 x^2+28 x+21$
• $y^2=47 x^6+4 x^5+62 x^4+47 x^3+45 x^2+8 x+49$
• $y^2=x^6+41 x^5+24 x^4+50 x^3+82 x^2+33 x+25$
• $y^2=18 x^6+70 x^5+80 x^4+29 x^3+80 x^2+70 x+18$
• $y^2=26 x^6+87 x^5+17 x^4+86 x^3+81 x^2+67 x+12$
• $y^2=73 x^6+77 x^5+18 x^4+53 x^3+28 x+22$
• $y^2=5 x^6+88 x^5+48 x^4+45 x^3+35 x^2+78 x+16$
• $y^2=34 x^6+x^5+86 x^4+4 x^3+58 x^2+87 x+39$
• $y^2=45 x^6+50 x^5+17 x^4+5 x^3+34 x^2+22 x+4$
• $y^2=88 x^6+2 x^5+46 x^4+75 x^3+86 x^2+65 x+2$
• $y^2=13 x^6+48 x^4+48 x^2+13$
• $y^2=74 x^6+85 x^5+36 x^4+66 x^3+81 x^2+2 x+28$
• $y^2=31 x^6+18 x^5+61 x^4+66 x^3+23 x^2+67 x+19$
• $y^2=22 x^6+54 x^5+5 x^4+87 x^3+81 x^2+35 x+53$
• $y^2=77 x^6+71 x^5+6 x^4+32 x^3+77 x^2+26 x+20$
• $y^2=81 x^6+40 x^5+49 x^4+13 x^3+49 x^2+40 x+81$
• $y^2=18 x^6+60 x^5+80 x^4+67 x^3+86 x^2+36 x+1$
• $y^2=14 x^6+81 x^5+57 x^4+54 x^3+68 x^2+64 x+60$
• $y^2=50 x^6+53 x^5+32 x^4+56 x^3+32 x^2+53 x+50$
• $y^2=67 x^6+76 x^5+76 x^4+63 x^3+57 x^2+12 x+24$
• $y^2=31 x^6+68 x^5+32 x^4+7 x^3+24 x^2+21 x+20$
• $y^2=33 x^6+21 x^4+21 x^2+33$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.m^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.ay_mk	$2$	(not in LMFDB)
2.89.a_bi	$2$	(not in LMFDB)
2.89.am_cd	$3$	(not in LMFDB)