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

• Label: 2.89.a_gw
• Base field: $\F_{89}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{89}$
Dimension:	$2$
L-polynomial:	$( 1 + 89 x^{2} )^{2}$
	$1 + 178 x^{2} + 7921 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.5$
Angle rank:	$0$ (numerical)
Jacobians:	$209$

This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8100$	$65610000$	$496982700900$	$3934601256960000$	$31181719941134302500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$8278$	$704970$	$62710558$	$5584059450$	$496984110838$	$44231334895530$	$3936588554733118$	$350356403707485210$	$31181719952302421398$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 209 curves (of which all are hyperelliptic):

• $y^2=x^5+88$
• $y^2=3 x^5+86$
• $y^2=11 x^6+9 x^5+74 x^4+50 x^3+31 x^2+46 x+35$
• $y^2=33 x^6+27 x^5+44 x^4+61 x^3+4 x^2+49 x+16$
• $y^2=70 x^6+x^5+63 x^4+50 x^3+37 x^2+60 x+50$
• $y^2=32 x^6+3 x^5+11 x^4+61 x^3+22 x^2+2 x+61$
• $y^2=67 x^6+44 x^5+67 x^4+27 x^3+74 x^2+x$
• $y^2=53 x^6+15 x^5+24 x^4+58 x^3+71 x^2+32 x+20$
• $y^2=70 x^6+45 x^5+72 x^4+85 x^3+35 x^2+7 x+60$
• $y^2=46 x^6+61 x^5+46 x^4+49 x^2+74 x+85$
• $y^2=79 x^6+15 x^5+29 x^4+17 x^3+29 x^2+15 x+79$
• $y^2=59 x^6+45 x^5+87 x^4+51 x^3+87 x^2+45 x+59$
• $y^2=27 x^6+50 x^5+44 x^4+65 x^3+44 x^2+50 x+27$
• $y^2=81 x^6+61 x^5+43 x^4+17 x^3+43 x^2+61 x+81$
• $y^2=2 x^6+39 x^5+74 x^4+38 x^3+80 x^2+5 x+55$
• $y^2=6 x^6+28 x^5+44 x^4+25 x^3+62 x^2+15 x+76$
• $y^2=18 x^6+76 x^5+53 x^4+34 x^3+45 x^2+52 x+1$
• $y^2=54 x^6+50 x^5+70 x^4+13 x^3+46 x^2+67 x+3$
• $y^2=23 x^6+31 x^5+84 x^4+69 x^3+84 x^2+31 x+23$
• $y^2=69 x^6+4 x^5+74 x^4+29 x^3+74 x^2+4 x+69$
• and

• $y^2=42 x^6+65 x^4+65 x^2+42$
• $y^2=37 x^6+17 x^4+17 x^2+37$
• $y^2=42 x^6+17 x^4+51 x^2+66$
• $y^2=49 x^6+x^4+3 x^2+77$
• $y^2=41 x^6+74 x^5+36 x^4+24 x^3+52 x^2+13 x+65$
• $y^2=67 x^6+53 x^5+3 x^4+18 x^3+19 x^2+69 x+5$
• $y^2=23 x^6+70 x^5+9 x^4+54 x^3+57 x^2+29 x+15$
• $y^2=4 x^6+15 x^5+29 x^3+15 x+4$
• $y^2=12 x^6+45 x^5+87 x^3+45 x+12$
• $y^2=x^6+x^3+50$
• $y^2=3 x^6+3 x^3+61$
• $y^2=34 x^6+55 x^5+62 x^4+27 x^3+63 x^2+72 x+79$
• $y^2=13 x^6+76 x^5+8 x^4+81 x^3+11 x^2+38 x+59$
• $y^2=x^6+x^3+84$
• $y^2=3 x^6+3 x^3+74$
• $y^2=51 x^6+20 x^5+28 x^4+4 x^3+58 x^2+84 x+61$
• $y^2=64 x^6+60 x^5+84 x^4+12 x^3+85 x^2+74 x+5$
• $y^2=88 x^6+48 x^5+75 x^4+50 x^3+85 x^2+85 x+34$
• $y^2=86 x^6+55 x^5+47 x^4+61 x^3+77 x^2+77 x+13$
• $y^2=18 x^6+60 x^5+73 x^4+17 x^3+49 x^2+19 x+81$
• $y^2=54 x^6+2 x^5+41 x^4+51 x^3+58 x^2+57 x+65$
• $y^2=76 x^6+71 x^5+52 x^4+39 x^3+52 x^2+71 x+76$
• $y^2=50 x^6+35 x^5+67 x^4+28 x^3+67 x^2+35 x+50$
• $y^2=x^6+x^3+42$
• $y^2=3 x^6+3 x^3+37$
• $y^2=50 x^6+75 x^5+71 x^4+68 x^3+x^2+6 x+69$
• $y^2=61 x^6+47 x^5+35 x^4+26 x^3+3 x^2+18 x+29$
• $y^2=x^6+x^3+51$
• $y^2=88 x^6+48 x^5+10 x^4+22 x^3+6 x^2+60 x+66$
• $y^2=82 x^6+7 x^5+7 x^4+11 x^3+x^2+72 x+12$
• $y^2=68 x^6+21 x^5+21 x^4+33 x^3+3 x^2+38 x+36$
• $y^2=50 x^6+12 x^5+22 x^4+31 x^3+11 x^2+21 x+31$
• $y^2=61 x^6+36 x^5+66 x^4+4 x^3+33 x^2+63 x+4$
• $y^2=55 x^6+2 x^5+32 x^4+50 x^3+43 x^2+80 x+45$
• $y^2=76 x^6+6 x^5+7 x^4+61 x^3+40 x^2+62 x+46$
• $y^2=29 x^6+7 x^5+59 x^4+25 x^3+24 x^2+86 x+6$
• $y^2=87 x^6+21 x^5+88 x^4+75 x^3+72 x^2+80 x+18$
• $y^2=20 x^6+30 x^5+54 x^4+71 x^3+41 x^2+59 x+59$
• $y^2=60 x^6+x^5+73 x^4+35 x^3+34 x^2+88 x+88$
• $y^2=14 x^6+72 x^5+72 x^4+38 x^3+72 x^2+72 x+14$
• $y^2=42 x^6+38 x^5+38 x^4+25 x^3+38 x^2+38 x+42$
• $y^2=10 x^6+12 x^5+43 x^4+68 x^3+43 x^2+12 x+10$
• $y^2=30 x^6+36 x^5+40 x^4+26 x^3+40 x^2+36 x+30$
• $y^2=87 x^6+70 x^5+82 x^4+8 x^3+15 x^2+29 x+49$
• $y^2=83 x^6+32 x^5+68 x^4+24 x^3+45 x^2+87 x+58$
• $y^2=38 x^6+45 x^5+40 x^4+25 x^3+40 x^2+45 x+38$
• $y^2=25 x^6+46 x^5+31 x^4+75 x^3+31 x^2+46 x+25$
• $y^2=x^6+88$
• $y^2=3 x^6+86$
• $y^2=x^6+62$
• $y^2=x^6+67 x^5+30 x^4+56 x^3+59 x^2+67 x+88$
• $y^2=12 x^6+16 x^5+37 x^4+31 x^3+38 x^2+67 x+48$
• $y^2=36 x^6+48 x^5+22 x^4+4 x^3+25 x^2+23 x+55$
• $y^2=53 x^6+10 x^5+25 x^4+75 x^3+25 x^2+10 x+53$
• $y^2=70 x^6+30 x^5+75 x^4+47 x^3+75 x^2+30 x+70$
• $y^2=5 x^6+62 x^5+3 x^4+62 x^3+83 x^2+70 x+49$
• $y^2=15 x^6+8 x^5+9 x^4+8 x^3+71 x^2+32 x+58$
• $y^2=42 x^6+8 x^5+70 x^4+45 x^3+70 x^2+8 x+42$
• $y^2=37 x^6+24 x^5+32 x^4+46 x^3+32 x^2+24 x+37$
• $y^2=4 x^6+60 x^5+19 x^4+41 x^3+77 x^2+37 x+5$
• $y^2=12 x^6+2 x^5+57 x^4+34 x^3+53 x^2+22 x+15$
• $y^2=27 x^6+75 x^5+4 x^4+39 x^3+4 x^2+75 x+27$
• $y^2=81 x^6+47 x^5+12 x^4+28 x^3+12 x^2+47 x+81$
• $y^2=51 x^6+19 x^5+76 x^4+27 x^3+11 x^2+86 x+86$
• $y^2=64 x^6+57 x^5+50 x^4+81 x^3+33 x^2+80 x+80$
• $y^2=77 x^6+79 x^5+3 x^4+26 x^3+6 x^2+49 x+82$
• $y^2=53 x^6+59 x^5+9 x^4+78 x^3+18 x^2+58 x+68$
• $y^2=87 x^6+38 x^5+5 x^4+54 x^3+5 x^2+38 x+87$
• $y^2=83 x^6+25 x^5+15 x^4+73 x^3+15 x^2+25 x+83$
• $y^2=73 x^6+44 x^5+80 x^4+29 x^3+32 x^2+8 x+12$
• $y^2=41 x^6+43 x^5+62 x^4+87 x^3+7 x^2+24 x+36$
• $y^2=x^6+60 x^5+82 x^4+29 x^3+11 x^2+64 x+24$
• $y^2=18 x^6+47 x^5+47 x^4+43 x^3+38 x^2+74 x+22$
• $y^2=54 x^6+52 x^5+52 x^4+40 x^3+25 x^2+44 x+66$
• $y^2=83 x^6+4 x^5+3 x^4+50 x^3+19 x^2+22 x+58$
• $y^2=71 x^6+12 x^5+9 x^4+61 x^3+57 x^2+66 x+85$
• $y^2=50 x^6+44 x^5+21 x^4+21 x^3+21 x^2+44 x+50$
• $y^2=61 x^6+43 x^5+63 x^4+63 x^3+63 x^2+43 x+61$
• $y^2=23 x^6+10 x^5+25 x^4+15 x^3+88 x^2+5 x+12$
• $y^2=69 x^6+30 x^5+75 x^4+45 x^3+86 x^2+15 x+36$
• $y^2=44 x^6+3 x^5+6 x^4+17 x^3+6 x^2+3 x+44$
• $y^2=43 x^6+9 x^5+18 x^4+51 x^3+18 x^2+9 x+43$
• $y^2=25 x^6+87 x^5+82 x^4+14 x^3+49 x^2+66 x+28$
• $y^2=75 x^6+83 x^5+68 x^4+42 x^3+58 x^2+20 x+84$
• $y^2=50 x^6+8 x^5+71 x^4+76 x^3+68 x^2+x+11$
• $y^2=61 x^6+24 x^5+35 x^4+50 x^3+26 x^2+3 x+33$
• $y^2=85 x^6+75 x^5+85 x^4+27 x^3+85 x^2+75 x+85$
• $y^2=77 x^6+47 x^5+77 x^4+81 x^3+77 x^2+47 x+77$
• $y^2=61 x^6+77 x^5+76 x^4+4 x^3+12 x^2+14 x+52$
• $y^2=5 x^6+53 x^5+50 x^4+12 x^3+36 x^2+42 x+67$
• $y^2=54 x^6+51 x^5+86 x^4+24 x^3+86 x^2+51 x+54$
• $y^2=73 x^6+64 x^5+80 x^4+72 x^3+80 x^2+64 x+73$
• $y^2=42 x^6+55 x^5+x^4+20 x^3+24 x^2+62 x+31$
• $y^2=37 x^6+76 x^5+3 x^4+60 x^3+72 x^2+8 x+4$
• $y^2=55 x^6+50 x^5+51 x^4+53 x^3+33 x^2+75 x+65$
• $y^2=76 x^6+61 x^5+64 x^4+70 x^3+10 x^2+47 x+17$
• $y^2=54 x^6+5 x^4+5 x^2+54$
• $y^2=73 x^6+15 x^4+15 x^2+73$
• $y^2=54 x^6+15 x^4+45 x^2+34$
• $y^2=37 x^6+3 x^4+9 x^2+20$
• $y^2=37 x^6+14 x^5+49 x^4+5 x^3+44 x^2+65 x+24$
• $y^2=22 x^6+42 x^5+58 x^4+15 x^3+43 x^2+17 x+72$
• $y^2=44 x^6+70 x^5+17 x^4+x^3+17 x^2+70 x+44$
• $y^2=43 x^6+32 x^5+51 x^4+3 x^3+51 x^2+32 x+43$
• $y^2=54 x^6+16 x^5+66 x^4+30 x^3+74 x^2+38 x+29$
• $y^2=73 x^6+48 x^5+20 x^4+x^3+44 x^2+25 x+87$
• $y^2=11 x^6+85 x^5+76 x^4+83 x^3+14 x^2+8 x+10$
• $y^2=33 x^6+77 x^5+50 x^4+71 x^3+42 x^2+24 x+30$
• $y^2=68 x^6+68 x^5+82 x^4+29 x^3+82 x^2+68 x+68$
• $y^2=26 x^6+26 x^5+68 x^4+87 x^3+68 x^2+26 x+26$
• $y^2=46 x^6+62 x^5+78 x^4+81 x^3+36 x^2+19 x+52$
• $y^2=49 x^6+8 x^5+56 x^4+65 x^3+19 x^2+57 x+67$
• $y^2=7 x^6+81 x^5+7 x^4+85 x^3+7 x^2+81 x+7$
• $y^2=21 x^6+65 x^5+21 x^4+77 x^3+21 x^2+65 x+21$
• $y^2=21 x^6+59 x^5+11 x^4+53 x^3+21 x^2+26 x+85$
• $y^2=63 x^6+88 x^5+33 x^4+70 x^3+63 x^2+78 x+77$
• $y^2=88 x^6+28 x^5+85 x^4+85 x^3+85 x^2+28 x+88$
• $y^2=86 x^6+84 x^5+77 x^4+77 x^3+77 x^2+84 x+86$
• $y^2=15 x^6+51 x^5+33 x^4+42 x^3+48 x^2+60 x+43$
• $y^2=41 x^6+26 x^5+32 x^4+45 x^3+21 x^2+53 x+15$
• $y^2=34 x^6+78 x^5+7 x^4+46 x^3+63 x^2+70 x+45$
• $y^2=3 x^6+81 x^5+23 x^4+17 x^3+7 x^2+6 x+75$
• $y^2=x^6+x^3+49$
• $y^2=3 x^6+3 x^3+58$
• $y^2=69 x^6+45 x^5+22 x^4+77 x^3+80 x^2+25 x+45$
• $y^2=29 x^6+46 x^5+66 x^4+53 x^3+62 x^2+75 x+46$
• $y^2=28 x^6+74 x^5+67 x^4+20 x^3+x^2+48 x+50$
• $y^2=84 x^6+44 x^5+23 x^4+60 x^3+3 x^2+55 x+61$
• $y^2=44 x^6+72 x^5+19 x^4+35 x^3+66 x+6$
• $y^2=43 x^6+38 x^5+57 x^4+16 x^3+20 x+18$
• $y^2=33 x^6+66 x^5+63 x^4+33 x^3+58 x^2+33 x+83$
• $y^2=10 x^6+20 x^5+11 x^4+10 x^3+85 x^2+10 x+71$
• $y^2=44 x^6+87 x^5+17 x^4+10 x^3+17 x^2+87 x+44$
• $y^2=43 x^6+83 x^5+51 x^4+30 x^3+51 x^2+83 x+43$
• $y^2=52 x^6+74 x^5+77 x^4+69 x^3+17 x^2+75 x+60$
• $y^2=67 x^6+44 x^5+53 x^4+29 x^3+51 x^2+47 x+2$
• $y^2=32 x^6+54 x^5+49 x^4+82 x^3+80 x^2+86 x+52$
• $y^2=72 x^6+38 x^5+66 x^4+12 x^3+22 x^2+13 x+48$
• $y^2=38 x^6+25 x^5+20 x^4+36 x^3+66 x^2+39 x+55$
• $y^2=59 x^6+12 x^5+82 x^4+43 x^3+34 x^2+73 x+22$
• $y^2=88 x^6+36 x^5+68 x^4+40 x^3+13 x^2+41 x+66$
• $y^2=10 x^6+73 x^5+52 x^4+67 x^2+55 x+3$
• $y^2=58 x^6+62 x^5+81 x^4+29 x^3+11 x^2+31 x+38$
• $y^2=85 x^6+8 x^5+65 x^4+87 x^3+33 x^2+4 x+25$
• $y^2=68 x^6+62 x^5+41 x^4+36 x^3+41 x^2+62 x+68$
• $y^2=26 x^6+8 x^5+34 x^4+19 x^3+34 x^2+8 x+26$
• $y^2=47 x^6+70 x^4+8 x^3+70 x^2+47$
• $y^2=52 x^6+32 x^4+24 x^3+32 x^2+52$
• $y^2=53 x^6+14 x^5+55 x^4+6 x^3+55 x^2+14 x+53$
• $y^2=70 x^6+42 x^5+76 x^4+18 x^3+76 x^2+42 x+70$
• $y^2=23 x^6+39 x^5+42 x^4+55 x^3+68 x^2+32 x+75$
• $y^2=69 x^6+28 x^5+37 x^4+76 x^3+26 x^2+7 x+47$
• $y^2=87 x^6+44 x^5+63 x^4+47 x^3+63 x^2+44 x+87$
• $y^2=83 x^6+43 x^5+11 x^4+52 x^3+11 x^2+43 x+83$
• $y^2=24 x^6+7 x^5+4 x^4+12 x^3+25 x^2+18 x+40$
• $y^2=72 x^6+21 x^5+12 x^4+36 x^3+75 x^2+54 x+31$
• $y^2=3 x^6+60 x^5+83 x^4+59 x^3+28 x^2+31 x+61$
• $y^2=9 x^6+2 x^5+71 x^4+88 x^3+84 x^2+4 x+5$
• $y^2=24 x^6+61 x^5+11 x^4+67 x^3+11 x^2+61 x+24$
• $y^2=72 x^6+5 x^5+33 x^4+23 x^3+33 x^2+5 x+72$
• $y^2=x^6+28 x^5+78 x^4+57 x^3+54 x^2+77 x+61$
• $y^2=3 x^6+84 x^5+56 x^4+82 x^3+73 x^2+53 x+5$
• $y^2=6 x^6+49 x^5+13 x^4+51 x^3+39 x^2+10 x+37$
• $y^2=18 x^6+58 x^5+39 x^4+64 x^3+28 x^2+30 x+22$
• $y^2=x^6+x^3+45$
• $y^2=3 x^6+3 x^3+46$
• $y^2=61 x^6+59 x^5+65 x^4+77 x^3+61 x^2+63 x+33$
• $y^2=5 x^6+88 x^5+17 x^4+53 x^3+5 x^2+11 x+10$
• $y^2=54 x^6+72 x^5+38 x^4+6 x^3+38 x^2+72 x+54$
• $y^2=73 x^6+38 x^5+25 x^4+18 x^3+25 x^2+38 x+73$
• $y^2=73 x^6+57 x^5+68 x^4+2 x^3+68 x^2+57 x+73$
• $y^2=41 x^6+82 x^5+26 x^4+6 x^3+26 x^2+82 x+41$
• $y^2=7 x^6+51 x^5+21 x^4+2 x^3+52 x^2+40 x+19$
• $y^2=21 x^6+64 x^5+63 x^4+6 x^3+67 x^2+31 x+57$
• $y^2=15 x^6+30 x^5+22 x^4+74 x^3+78 x^2+52 x+76$
• $y^2=45 x^6+x^5+66 x^4+44 x^3+56 x^2+67 x+50$
• $y^2=28 x^6+47 x^5+72 x^4+63 x^3+72 x^2+47 x+28$
• $y^2=84 x^6+52 x^5+38 x^4+11 x^3+38 x^2+52 x+84$
• $y^2=5 x^5+72 x^4+12 x^3+22 x^2+67 x+30$
• $y^2=15 x^5+38 x^4+36 x^3+66 x^2+23 x+1$
• $y^2=83 x^6+63 x^5+30 x^4+23 x^3+39 x^2+50 x+74$
• $y^2=71 x^6+11 x^5+x^4+69 x^3+28 x^2+61 x+44$
• $y^2=33 x^6+48 x^5+79 x^4+62 x^3+16 x^2+10 x+26$
• $y^2=45 x^6+79 x^5+26 x^4+87 x^3+38 x^2+20 x+88$
• $y^2=46 x^6+59 x^5+78 x^4+83 x^3+25 x^2+60 x+86$
• $y^2=88 x^6+70 x^5+12 x^4+46 x^3+14 x^2+2 x+61$
• $y^2=86 x^6+32 x^5+36 x^4+49 x^3+42 x^2+6 x+5$
• $y^2=82 x^6+27 x^5+74 x^4+48 x^3+8 x^2+29 x+48$
• $y^2=68 x^6+81 x^5+44 x^4+55 x^3+24 x^2+87 x+55$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{89^{2}}$.

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-89}) \)$)$

Endomorphism algebra over $\overline{\F}_{89}$

The base change of $A$ to $\F_{89^{2}}$ is 1.7921.gw^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $89$ and $\infty$.

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_adl	$3$	(not in LMFDB)
2.89.a_agw	$4$	(not in LMFDB)
2.89.a_a	$8$	(not in LMFDB)