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

• Label: 2.89.ak_hv
• 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:	$( 1 - 5 x + 89 x^{2} )^{2}$
	$1 - 10 x + 203 x^{2} - 890 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.414628214971$, $\pm0.414628214971$
Angle rank:	$1$ (numerical)
Jacobians:	$45$
Cyclic group of points:	no
Non-cyclic primes:	$5, 17$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7225$	$65205625$	$498690192400$	$3935639447355625$	$31180094721808905625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$80$	$8228$	$707390$	$62727108$	$5583768400$	$496981182638$	$44231360257360$	$3936588942152068$	$350356402132532270$	$31181719909947370148$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 45 curves (of which all are hyperelliptic):

• $y^2=65 x^6+12 x^5+27 x^4+43 x^3+27 x^2+81 x+55$
• $y^2=25 x^6+73 x^5+71 x^4+43 x^3+15 x^2+80 x+38$
• $y^2=88 x^6+87 x^5+38 x^4+61 x^3+85 x^2+45 x+38$
• $y^2=41 x^6+85 x^5+37 x^4+29 x^3+13 x^2+22 x+23$
• $y^2=42 x^6+20 x^5+25 x^4+87 x^3+72 x^2+42 x+81$
• $y^2=2 x^6+70 x^5+8 x^4+65 x^3+38 x^2+56 x+34$
• $y^2=54 x^6+66 x^5+16 x^4+15 x^3+71 x^2+65 x+55$
• $y^2=41 x^6+7 x^5+10 x^4+37 x^3+4 x^2+83 x+19$
• $y^2=78 x^6+82 x^5+21 x^4+63 x^3+84 x^2+27 x+17$
• $y^2=22 x^6+88 x^5+66 x^4+68 x^3+80 x^2+55 x+73$
• $y^2=52 x^6+8 x^5+38 x^4+46 x^3+30 x^2+8 x+29$
• $y^2=79 x^6+8 x^5+46 x^4+50 x^3+62 x^2+46 x+13$
• $y^2=72 x^6+17 x^4+8 x^3+24 x^2+30 x+84$
• $y^2=83 x^6+34 x^5+24 x^4+58 x^3+85 x^2+13 x+35$
• $y^2=9 x^6+72 x^5+14 x^4+37 x^3+49 x^2+33 x+7$
• $y^2=76 x^6+33 x^5+56 x^4+64 x^3+7 x^2+77 x+37$
• $y^2=26 x^6+88 x^5+16 x^4+64 x^3+50 x^2+25 x+38$
• $y^2=30 x^6+66 x^5+59 x^4+45 x^3+74 x^2+38 x+7$
• $y^2=2 x^6+77 x^5+x^4+26 x^3+16 x^2+43 x+4$
• $y^2=86 x^6+80 x^5+21 x^4+88 x^3+73 x^2+82 x+12$
• and

• $y^2=67 x^6+8 x^5+58 x^4+15 x^3+79 x^2+72 x+43$
• $y^2=19 x^6+42 x^5+81 x^4+62 x^3+41 x^2+38 x+68$
• $y^2=88 x^6+60 x^5+57 x^4+30 x^2+69 x+37$
• $y^2=9 x^6+73 x^5+20 x^4+54 x^3+42 x^2+22 x+10$
• $y^2=27 x^6+6 x^4+45 x^3+66 x^2+32 x+60$
• $y^2=42 x^6+37 x^5+13 x^4+88 x^3+60 x^2+82 x+2$
• $y^2=88 x^6+71 x^5+40 x^4+8 x^3+25 x^2+10 x+31$
• $y^2=30 x^6+28 x^5+38 x^4+50 x^3+13 x^2+48 x+6$
• $y^2=32 x^6+60 x^5+11 x^4+55 x^3+25 x^2+26 x+45$
• $y^2=85 x^6+27 x^5+32 x^4+40 x^3+4 x^2+63 x+73$
• $y^2=24 x^6+54 x^5+10 x^4+64 x^3+79 x^2+86 x+76$
• $y^2=31 x^6+64 x^5+35 x^4+74 x^3+55 x^2+77 x+66$
• $y^2=80 x^6+56 x^5+11 x^4+61 x^3+28 x^2+65 x+58$
• $y^2=86 x^6+53 x^5+28 x^3+78 x^2+41 x+10$
• $y^2=74 x^6+51 x^5+25 x^4+41 x^3+17 x^2+27 x+48$
• $y^2=27 x^6+63 x^5+16 x^4+58 x^3+12 x^2+14 x+15$
• $y^2=65 x^6+88 x^5+70 x^4+4 x^3+88 x^2+61 x+72$
• $y^2=7 x^6+21 x^5+60 x^4+48 x^3+81 x^2+86 x+84$
• $y^2=16 x^6+62 x^5+60 x^4+43 x^3+60 x^2+62 x+16$
• $y^2=81 x^6+25 x^5+33 x^4+56 x^3+47 x^2+68 x+37$
• $y^2=19 x^6+16 x^5+50 x^4+86 x^3+21 x^2+81 x+77$
• $y^2=43 x^6+7 x^5+79 x^4+40 x^3+57 x^2+53 x+84$
• $y^2=14 x^6+12 x^5+64 x^4+15 x^3+33 x^2+5 x+13$
• $y^2=58 x^6+11 x^5+25 x^4+2 x^3+72 x^2+58 x+46$
• $y^2=45 x^6+6 x^5+73 x^4+28 x^3+79 x^2+83 x+72$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-331}) \)$)$

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_fx	$2$	(not in LMFDB)
2.89.k_hv	$2$	(not in LMFDB)
2.89.f_acm	$3$	(not in LMFDB)