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

• Label: 2.89.u_kq
• Base field: $\F_{89}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{89}$
Dimension:	$2$
L-polynomial:	$1 + 20 x + 276 x^{2} + 1780 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.650377107003$, $\pm0.706807754149$
Angle rank:	$2$ (numerical)
Number field:	4.0.4077824.3
Galois group:	$D_{4}$
Jacobians:	$52$
Cyclic group of points:	yes

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9998$	$63967204$	$494715287150$	$3937751737790864$	$31182217479155572078$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$110$	$8074$	$701750$	$62760774$	$5584148550$	$496978921738$	$44231350207070$	$3936588847987134$	$350356402147256750$	$31181719941012160554$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 52 curves (of which all are hyperelliptic):

• $y^2=55 x^6+42 x^5+9 x^4+52 x^3+74 x^2+11 x+7$
• $y^2=29 x^6+41 x^5+64 x^3+52 x^2+32 x+19$
• $y^2=5 x^6+66 x^5+6 x^4+38 x^3+30 x^2+57 x+35$
• $y^2=69 x^6+49 x^5+78 x^4+80 x^3+72 x^2+30 x+45$
• $y^2=63 x^6+19 x^5+44 x^4+4 x^3+7 x^2+38 x+74$
• $y^2=84 x^6+74 x^5+49 x^4+48 x^3+61 x^2+30 x+6$
• $y^2=17 x^6+20 x^5+63 x^4+22 x^3+52 x^2+45 x+36$
• $y^2=38 x^6+53 x^5+41 x^4+34 x^3+30 x^2+76 x+12$
• $y^2=25 x^6+73 x^5+62 x^4+27 x^3+20 x^2+77 x+80$
• $y^2=48 x^6+85 x^5+43 x^4+7 x^3+87 x^2+59 x+80$
• $y^2=2 x^6+43 x^5+74 x^4+9 x^3+85 x^2+24 x+50$
• $y^2=41 x^6+71 x^5+44 x^4+7 x^3+6 x^2+40 x+74$
• $y^2=52 x^6+56 x^5+47 x^4+65 x^3+9 x^2+4 x+53$
• $y^2=54 x^6+20 x^5+14 x^4+2 x^3+60 x^2+16 x+32$
• $y^2=64 x^6+34 x^5+51 x^4+88 x^3+63 x^2+60 x+24$
• $y^2=70 x^6+42 x^5+44 x^4+5 x^3+81 x^2+60 x+2$
• $y^2=30 x^6+80 x^5+16 x^4+82 x^3+29 x^2+15 x$
• $y^2=68 x^6+10 x^5+8 x^4+80 x^3+71 x^2+68 x+53$
• $y^2=17 x^6+20 x^5+36 x^4+17 x^3+60 x^2+31 x+59$
• $y^2=2 x^6+2 x^5+45 x^4+47 x^3+62 x^2+81 x+38$
• and

• $y^2=9 x^6+80 x^5+16 x^4+31 x^3+64 x^2+42 x+8$
• $y^2=69 x^6+24 x^5+40 x^4+18 x^3+65 x^2+45 x+9$
• $y^2=42 x^6+44 x^5+7 x^4+64 x^3+36 x^2+45 x+69$
• $y^2=21 x^6+45 x^5+23 x^4+49 x^3+32 x^2+54 x+58$
• $y^2=16 x^6+80 x^5+88 x^4+50 x^3+41 x^2+21 x+65$
• $y^2=42 x^6+49 x^5+7 x^4+5 x^3+39 x^2+47 x+81$
• $y^2=6 x^6+7 x^5+24 x^4+29 x^3+87 x^2+39 x+50$
• $y^2=26 x^6+69 x^5+14 x^4+73 x^3+73 x^2+82 x+21$
• $y^2=64 x^6+58 x^5+38 x^4+49 x^3+52 x^2+33 x+19$
• $y^2=71 x^6+12 x^5+14 x^4+81 x^3+59 x^2+20 x+57$
• $y^2=63 x^6+33 x^5+60 x^4+85 x^3+80 x^2+6 x+50$
• $y^2=21 x^6+86 x^5+73 x^4+85 x^3+38 x^2+45 x+6$
• $y^2=61 x^6+64 x^5+18 x^4+56 x^3+58 x^2+87 x+29$
• $y^2=26 x^6+17 x^5+85 x^4+53 x^3+78 x^2+30 x+36$
• $y^2=4 x^6+4 x^5+41 x^4+67 x^3+44 x^2+26 x+59$
• $y^2=3 x^6+49 x^5+25 x^4+76 x^3+7 x^2+70 x+16$
• $y^2=18 x^6+40 x^5+75 x^4+10 x^3+83 x^2+39 x+67$
• $y^2=9 x^6+48 x^5+50 x^4+74 x^3+83 x^2+85 x+72$
• $y^2=22 x^6+80 x^5+30 x^4+31 x^3+16 x^2+20 x+24$
• $y^2=74 x^6+78 x^5+58 x^4+69 x^3+19 x^2+53 x+50$
• $y^2=62 x^6+76 x^5+59 x^4+31 x^3+58 x^2+55 x+88$
• $y^2=10 x^6+85 x^5+31 x^4+26 x^3+6 x^2+18 x+32$
• $y^2=85 x^6+72 x^5+76 x^4+22 x^3+62 x^2+2 x+1$
• $y^2=82 x^6+44 x^5+14 x^4+44 x^3+85 x^2+15 x+79$
• $y^2=81 x^6+34 x^5+48 x^4+75 x^3+49 x^2+40 x+6$
• $y^2=24 x^6+22 x^5+67 x^4+16 x^3+73 x^2+10 x+18$
• $y^2=32 x^6+49 x^5+35 x^4+4 x^3+51 x^2+81 x+31$
• $y^2=24 x^5+52 x^4+34 x^3+13 x^2+14 x+75$
• $y^2=11 x^6+64 x^5+43 x^4+57 x^3+82 x^2+58 x+59$
• $y^2=81 x^6+7 x^5+4 x^4+34 x^3+65 x^2+46 x+8$
• $y^2=79 x^6+74 x^5+60 x^4+32 x^3+27 x^2+68 x+55$
• $y^2=56 x^6+66 x^5+7 x^4+57 x^3+5 x^2+32 x+55$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The endomorphism algebra of this simple isogeny class is 4.0.4077824.3.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.89.au_kq	$2$	(not in LMFDB)