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

• Label: 2.89.bc_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:	$\pm0.766121877123$, $\pm0.766121877123$
Angle rank:	$1$ (numerical)
Jacobians:	$44$
Cyclic group of points:	no
Non-cyclic primes:	$2, 13$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10816$	$62473216$	$495582208576$	$3938536440217600$	$31180281660359480896$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$118$	$7886$	$702982$	$62773278$	$5583801878$	$496982134766$	$44231346006182$	$3936588575054398$	$350356405947704758$	$31181719919130753806$

Jacobians and polarizations

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

• $y^2=14 x^6+65 x^5+59 x^4+9 x^3+59 x^2+65 x+14$
• $y^2=36 x^6+14 x^5+63 x^4+37 x^3+38 x^2+12 x+84$
• $y^2=38 x^6+76 x^5+32 x^4+85 x^3+42 x^2+60 x+6$
• $y^2=22 x^6+2 x^5+51 x^4+21 x^3+67 x^2+83 x+55$
• $y^2=79 x^6+51 x^5+44 x^4+81 x^3+44 x^2+51 x+79$
• $y^2=84 x^6+80 x^5+28 x^4+33 x^3+40 x^2+76 x+38$
• $y^2=45 x^6+44 x^5+40 x^4+15 x^3+40 x^2+44 x+45$
• $y^2=x^6+18 x^5+63 x^4+11 x^3+33 x^2+34 x+17$
• $y^2=72 x^6+34 x^5+49 x^4+29 x^3+76 x^2+78 x+16$
• $y^2=11 x^6+14 x^5+33 x^4+14 x^3+10 x^2+30 x+46$
• $y^2=7 x^6+78 x^4+78 x^2+7$
• $y^2=46 x^6+10 x^5+87 x^4+x^3+54 x^2+26 x+10$
• $y^2=50 x^6+32 x^5+21 x^4+76 x^3+11 x^2+11 x+49$
• $y^2=32 x^6+5 x^5+76 x^4+45 x^3+80 x^2+59 x$
• $y^2=53 x^6+77 x^5+74 x^4+62 x^3+46 x^2+7 x+78$
• $y^2=60 x^6+29 x^5+27 x^4+48 x^3+48 x^2+45 x+53$
• $y^2=84 x^6+78 x^5+39 x^4+48 x^3+71 x^2+44 x+25$
• $y^2=61 x^6+23 x^5+23 x^4+13 x^3+74 x^2+33 x+76$
• $y^2=53 x^6+87 x^5+55 x^4+33 x^3+20 x^2+40 x+62$
• $y^2=84 x^6+32 x^5+51 x^4+37 x^3+26 x^2+21 x+81$
• and

• $y^2=7 x^6+85 x^5+35 x^4+22 x^3+27 x^2+44 x+47$
• $y^2=42 x^6+81 x^4+81 x^2+42$
• $y^2=36 x^6+61 x^5+37 x^4+70 x^3+19 x^2+87 x+30$
• $y^2=x^6+32 x^5+30 x^4+50 x^3+60 x^2+39 x+8$
• $y^2=40 x^6+69 x^5+53 x^3+84 x+5$
• $y^2=51 x^6+28 x^5+5 x^4+76 x^3+36 x^2+56 x+58$
• $y^2=50 x^6+82 x^5+35 x^4+67 x^3+39 x^2+49 x+80$
• $y^2=25 x^6+63 x^5+80 x^4+66 x^3+80 x^2+63 x+25$
• $y^2=53 x^6+65 x^5+83 x^4+68 x^3+83 x^2+65 x+53$
• $y^2=87 x^6+5 x^5+28 x^4+44 x^3+77 x^2+3 x+12$
• $y^2=50 x^6+11 x^5+70 x^4+31 x^3+70 x^2+11 x+50$
• $y^2=71 x^6+25 x^4+25 x^2+71$
• $y^2=37 x^6+45 x^5+80 x^4+11 x^3+15 x^2+50 x+48$
• $y^2=47 x^6+x^5+84 x^4+78 x^3+46 x^2+65 x+72$
• $y^2=45 x^6+18 x^5+55 x^4+5 x^3+31 x+73$
• $y^2=20 x^6+13 x^4+13 x^2+20$
• $y^2=34 x^6+21 x^5+49 x^4+68 x^3+49 x^2+21 x+34$
• $y^2=22 x^6+39 x^5+60 x^4+44 x^3+27 x^2+x+16$
• $y^2=5 x^6+8 x^5+59 x^4+20 x^3+8 x^2+33 x+36$
• $y^2=85 x^6+26 x^5+69 x^4+60 x^3+69 x^2+26 x+85$
• $y^2=55 x^6+88 x^5+87 x^4+18 x^3+87 x^2+88 x+55$
• $y^2=57 x^6+30 x^5+18 x^4+38 x^3+53 x^2+12 x+50$
• $y^2=34 x^6+32 x^4+32 x^2+34$
• $y^2=14 x^6+54 x^5+44 x^4+16 x^3+20 x^2+2 x+84$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.o^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.abc_ok	$2$	(not in LMFDB)
2.89.a_as	$2$	(not in LMFDB)
2.89.ao_ed	$3$	(not in LMFDB)