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

• Label: 2.89.bg_qo
• 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:	$( 1 + 14 x + 89 x^{2} )( 1 + 18 x + 89 x^{2} )$
	$1 + 32 x + 430 x^{2} + 2848 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.766121877123$, $\pm0.903075820349$
Angle rank:	$2$ (numerical)
Jacobians:	$44$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11232$	$61461504$	$497004240096$	$3937219029319680$	$31181041097265507552$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$122$	$7758$	$705002$	$62752286$	$5583937882$	$496982070126$	$44231333377738$	$3936588785898046$	$350356403737745978$	$31181719935664488078$

Jacobians and polarizations

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

• $y^2=63 x^6+45 x^5+18 x^4+52 x^3+47 x^2+67 x+60$
• $y^2=71 x^6+11 x^5+25 x^4+39 x^3+25 x^2+11 x+71$
• $y^2=78 x^6+20 x^5+29 x^4+6 x^3+19 x^2+75 x+57$
• $y^2=69 x^6+30 x^5+10 x^4+63 x^3+37 x^2+28 x+85$
• $y^2=36 x^6+39 x^5+85 x^4+65 x^3+25 x^2+59 x+21$
• $y^2=78 x^6+9 x^5+23 x^4+23 x^3+29 x^2+11 x+80$
• $y^2=88 x^6+12 x^5+30 x^4+55 x^3+30 x^2+12 x+88$
• $y^2=82 x^6+15 x^5+66 x^4+68 x^3+18 x^2+34 x+66$
• $y^2=44 x^6+17 x^5+18 x^4+53 x^3+12 x^2+70 x+19$
• $y^2=87 x^6+45 x^5+64 x^4+20 x^3+36 x^2+80 x+68$
• $y^2=68 x^6+28 x^5+48 x^4+12 x^3+8 x^2+88 x+20$
• $y^2=50 x^6+62 x^5+55 x^4+21 x^3+32 x^2+29 x+24$
• $y^2=41 x^6+11 x^5+26 x^4+76 x^3+26 x^2+11 x+41$
• $y^2=37 x^6+17 x^5+46 x^4+84 x^3+59 x^2+19 x+49$
• $y^2=45 x^6+21 x^5+52 x^4+80 x^3+27 x^2+55 x+81$
• $y^2=34 x^6+84 x^5+50 x^4+68 x^3+50 x^2+84 x+34$
• $y^2=10 x^6+33 x^5+51 x^4+8 x^3+51 x^2+33 x+10$
• $y^2=22 x^6+12 x^5+52 x^4+19 x^3+52 x^2+12 x+22$
• $y^2=10 x^6+82 x^5+22 x^4+30 x^3+22 x^2+82 x+10$
• $y^2=23 x^6+29 x^5+49 x^4+34 x^3+72 x^2+3 x+78$
• and

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

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 $\times$ 1.89.s and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.89.o : \(\Q(\sqrt{-10}) \).
• 1.89.s : \(\Q(\sqrt{-2}) \).

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.abg_qo	$2$	(not in LMFDB)
2.89.ae_acw	$2$	(not in LMFDB)
2.89.e_acw	$2$	(not in LMFDB)