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

• Label: 2.89.abi_rz
• 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:	$( 1 - 17 x + 89 x^{2} )^{2}$
	$1 - 34 x + 467 x^{2} - 3026 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.142835623920$, $\pm0.142835623920$
Angle rank:	$1$ (numerical)
Jacobians:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5329$	$61011721$	$496455523216$	$3937030774452169$	$31182760181908623649$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$7700$	$704222$	$62749284$	$5584245736$	$496983831086$	$44231361498184$	$3936589031876164$	$350356405184808398$	$31181719934951184500$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=45 x^6+79 x^5+3 x^4+43 x^3+15 x^2+52 x+29$
• $y^2=21 x^6+27 x^5+13 x^4+61 x^3+37 x^2+76 x+80$
• $y^2=88 x^6+46 x^5+27 x^4+64 x^3+82 x^2+5 x+71$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.ar^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$

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_aeh	$2$	(not in LMFDB)
2.89.bi_rz	$2$	(not in LMFDB)
2.89.r_hs	$3$	(not in LMFDB)