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

• Label: 2.89.o_it
• 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:	$( 1 + 7 x + 89 x^{2} )^{2}$
	$1 + 14 x + 227 x^{2} + 1246 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.620984762852$, $\pm0.620984762852$
Angle rank:	$1$ (numerical)
Jacobians:	$42$
Cyclic group of points:	no
Non-cyclic primes:	$97$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9409$	$64818601$	$494833461136$	$3936488669722249$	$31183299211319534449$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$104$	$8180$	$701918$	$62740644$	$5584342264$	$496979453486$	$44231322587416$	$3936589055394244$	$350356403055062222$	$31181719912310542100$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.h^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-307}) \)$)$

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.ao_it	$2$	(not in LMFDB)
2.89.a_ez	$2$	(not in LMFDB)
2.89.ah_abo	$3$	(not in LMFDB)