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

• Label: 2.89.abc_ok
• 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:	$( 1 - 14 x + 89 x^{2} )^{2}$
	$1 - 28 x + 374 x^{2} - 2492 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.233878122877$, $\pm0.233878122877$
Angle rank:	$1$ (numerical)
Jacobians:	$44$
Cyclic group of points:	no
Non-cyclic primes:	$2, 19$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5776$	$62473216$	$498385169296$	$3938536440217600$	$31183158255080791696$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$7886$	$706958$	$62773278$	$5584317022$	$496982134766$	$44231323784878$	$3936588575054398$	$350356401467265662$	$31181719919130753806$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 44 curves (of which all are hyperelliptic):

• $y^2=42 x^6+17 x^5+88 x^4+27 x^3+88 x^2+17 x+42$
• $y^2=12 x^6+64 x^5+21 x^4+42 x^3+72 x^2+4 x+28$
• $y^2=72 x^6+55 x^5+70 x^4+58 x^3+14 x^2+20 x+2$
• $y^2=37 x^6+60 x^5+17 x^4+7 x^3+52 x^2+87 x+48$
• $y^2=59 x^6+64 x^5+43 x^4+65 x^3+43 x^2+64 x+59$
• $y^2=74 x^6+62 x^5+84 x^4+10 x^3+31 x^2+50 x+25$
• $y^2=15 x^6+74 x^5+43 x^4+5 x^3+43 x^2+74 x+15$
• $y^2=30 x^6+6 x^5+21 x^4+63 x^3+11 x^2+41 x+65$
• $y^2=24 x^6+41 x^5+46 x^4+69 x^3+55 x^2+26 x+35$
• $y^2=33 x^6+42 x^5+10 x^4+42 x^3+30 x^2+x+49$
• $y^2=32 x^6+26 x^4+26 x^2+32$
• $y^2=45 x^6+33 x^5+29 x^4+30 x^3+18 x^2+68 x+33$
• $y^2=61 x^6+7 x^5+63 x^4+50 x^3+33 x^2+33 x+58$
• $y^2=7 x^6+15 x^5+50 x^4+46 x^3+62 x^2+88 x$
• $y^2=70 x^6+53 x^5+44 x^4+8 x^3+49 x^2+21 x+56$
• $y^2=2 x^6+87 x^5+81 x^4+55 x^3+55 x^2+46 x+70$
• $y^2=28 x^6+26 x^5+13 x^4+16 x^3+83 x^2+74 x+38$
• $y^2=5 x^6+69 x^5+69 x^4+39 x^3+44 x^2+10 x+50$
• $y^2=77 x^6+29 x^5+48 x^4+11 x^3+66 x^2+43 x+80$
• $y^2=74 x^6+7 x^5+64 x^4+22 x^3+78 x^2+63 x+65$
• and

• $y^2=21 x^6+77 x^5+16 x^4+66 x^3+81 x^2+43 x+52$
• $y^2=14 x^6+27 x^4+27 x^2+14$
• $y^2=12 x^6+50 x^5+42 x^4+53 x^3+36 x^2+29 x+10$
• $y^2=3 x^6+7 x^5+x^4+61 x^3+2 x^2+28 x+24$
• $y^2=31 x^6+29 x^5+70 x^3+74 x+15$
• $y^2=64 x^6+84 x^5+15 x^4+50 x^3+19 x^2+79 x+85$
• $y^2=76 x^6+57 x^5+71 x^4+52 x^3+13 x^2+46 x+86$
• $y^2=38 x^6+21 x^5+86 x^4+22 x^3+86 x^2+21 x+38$
• $y^2=70 x^6+17 x^5+71 x^4+26 x^3+71 x^2+17 x+70$
• $y^2=83 x^6+15 x^5+84 x^4+43 x^3+53 x^2+9 x+36$
• $y^2=76 x^6+63 x^5+53 x^4+40 x^3+53 x^2+63 x+76$
• $y^2=83 x^6+38 x^4+38 x^2+83$
• $y^2=22 x^6+46 x^5+62 x^4+33 x^3+45 x^2+61 x+55$
• $y^2=75 x^6+30 x^5+28 x^4+26 x^3+45 x^2+81 x+24$
• $y^2=15 x^6+6 x^5+48 x^4+61 x^3+40 x+54$
• $y^2=66 x^6+34 x^4+34 x^2+66$
• $y^2=13 x^6+63 x^5+58 x^4+26 x^3+58 x^2+63 x+13$
• $y^2=66 x^6+28 x^5+2 x^4+43 x^3+81 x^2+3 x+48$
• $y^2=15 x^6+24 x^5+88 x^4+60 x^3+24 x^2+10 x+19$
• $y^2=77 x^6+78 x^5+29 x^4+2 x^3+29 x^2+78 x+77$
• $y^2=48 x^6+59 x^5+29 x^4+6 x^3+29 x^2+59 x+48$
• $y^2=82 x^6+x^5+54 x^4+25 x^3+70 x^2+36 x+61$
• $y^2=41 x^6+70 x^4+70 x^2+41$
• $y^2=42 x^6+73 x^5+43 x^4+48 x^3+60 x^2+6 x+74$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.ao^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$

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_as	$2$	(not in LMFDB)
2.89.bc_ok	$2$	(not in LMFDB)
2.89.o_ed	$3$	(not in LMFDB)