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

• Label: 2.89.abk_ti
• 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:	$( 1 - 18 x + 89 x^{2} )^{2}$
	$1 - 36 x + 502 x^{2} - 3204 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.0969241796512$, $\pm0.0969241796512$
Angle rank:	$1$ (numerical)
Jacobians:	$6$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5184$	$60466176$	$495537155136$	$3935902059085824$	$31181639329704181824$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$7630$	$702918$	$62731294$	$5584045014$	$496982005486$	$44231349041766$	$3936588996741694$	$350356405887183222$	$31181719952198222350$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=26 x^6+72 x^4+72 x^2+26$
• $y^2=38 x^6+25 x^5+37 x^4+67 x^3+52 x^2+25 x+51$
• $y^2=74 x^6+10 x^4+10 x^2+74$
• $y^2=53 x^6+55 x^5+74 x^4+80 x^3+14 x^2+40 x+39$
• $y^2=19 x^6+16 x^5+16 x^4+25 x^3+16 x^2+16 x+19$
• $y^2=x^5+88 x$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

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

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_afq	$2$	(not in LMFDB)
2.89.bk_ti	$2$	(not in LMFDB)
2.89.s_jb	$3$	(not in LMFDB)