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

• Label: 2.89.ba_nj
• 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:	$( 1 + 13 x + 89 x^{2} )^{2}$
	$1 + 26 x + 347 x^{2} + 2314 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.741949407251$, $\pm0.741949407251$
Angle rank:	$1$ (numerical)
Jacobians:	$17$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10609$	$62900761$	$495188060416$	$3938566940548009$	$31180697988451651249$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$116$	$7940$	$702422$	$62773764$	$5583876436$	$496980864686$	$44231356725364$	$3936588559852804$	$350356404960670598$	$31181719935555359300$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 17 curves (of which all are hyperelliptic):

• $y^2=59 x^6+57 x^5+3 x^4+46 x^3+38 x^2+8 x+6$
• $y^2=24 x^6+25 x^5+6 x^4+6 x^3+80 x^2+19 x+47$
• $y^2=72 x^6+74 x^5+47 x^4+24 x^3+69 x^2+72 x+61$
• $y^2=46 x^6+55 x^5+36 x^4+35 x^3+51 x^2+41 x+50$
• $y^2=20 x^6+72 x^5+56 x^4+8 x^3+49 x^2+69 x+68$
• $y^2=30 x^6+57 x^5+50 x^4+50 x^3+4 x^2+76 x+9$
• $y^2=28 x^6+2 x^5+72 x^4+70 x^3+5 x^2+67 x+33$
• $y^2=79 x^6+30 x^5+2 x^4+23 x^3+65 x^2+47 x+68$
• $y^2=81 x^6+88 x^5+50 x^4+37 x^3+63 x^2+87 x+43$
• $y^2=88 x^6+78 x^5+21 x^4+32 x^3+25 x^2+50 x+79$
• $y^2=16 x^6+23 x^5+53 x^4+47 x^3+20 x^2+6 x+8$
• $y^2=39 x^6+84 x^5+56 x^4+87 x^3+41 x^2+49 x+25$
• $y^2=7 x^6+79 x^5+28 x^4+52 x^3+86 x^2+21 x+37$
• $y^2=80 x^6+48 x^5+51 x^4+56 x^3+61 x^2+45 x+30$
• $y^2=80 x^6+30 x^5+19 x^4+61 x^3+16 x^2+69 x+30$
• $y^2=57 x^6+49 x^5+74 x^4+4 x^3+49 x^2+84 x+32$
• $y^2=71 x^6+29 x^5+74 x^3+85 x^2+3 x+6$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.n^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-187}) \)$)$

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.aba_nj	$2$	(not in LMFDB)
2.89.a_j	$2$	(not in LMFDB)
2.89.an_dc	$3$	(not in LMFDB)