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

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

Invariants

Base field:	$\F_{89}$
Dimension:	$2$
L-polynomial:	$1 - 126 x^{2} + 7921 x^{4}$
Frobenius angles:	$\pm0.124829211204$, $\pm0.875170788796$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-13}, \sqrt{19})\)
Galois group:	$C_2^2$
Jacobians:	$72$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7796$	$60777616$	$496982284724$	$3936584664715264$	$31181719937905493876$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$7670$	$704970$	$62742174$	$5584059450$	$496983278486$	$44231334895530$	$3936589056668734$	$350356403707485210$	$31181719945844804150$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 72 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-13}, \sqrt{19})\).

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

The base change of $A$ to $\F_{89^{2}}$ is 1.7921.aew^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-247}) \)$)$

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.a_ew	$4$	(not in LMFDB)