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

• Label: 2.89.be_pm
• 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:	$( 1 + 14 x + 89 x^{2} )( 1 + 16 x + 89 x^{2} )$
	$1 + 30 x + 402 x^{2} + 2670 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.766121877123$, $\pm0.822192315511$
Angle rank:	$2$ (numerical)
Jacobians:	$20$
Isomorphism classes:	68
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11024$	$61998976$	$496158060944$	$3938174955520000$	$31180216451217186704$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$120$	$7826$	$703800$	$62767518$	$5583790200$	$496983091826$	$44231330888760$	$3936588720644158$	$350356405194366840$	$31181719915986452306$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=81 x^6+12 x^5+35 x^4+60 x^3+35 x^2+12 x+81$
• $y^2=50 x^6+62 x^5+25 x^4+3 x^3+25 x^2+62 x+50$
• $y^2=71 x^6+68 x^5+81 x^4+34 x^3+10 x^2+84 x+88$
• $y^2=32 x^6+29 x^5+76 x^4+75 x^3+24 x^2+13 x+84$
• $y^2=24 x^6+42 x^5+57 x^4+37 x^3+57 x^2+42 x+24$
• $y^2=44 x^6+15 x^5+3 x^4+28 x^3+3 x^2+15 x+44$
• $y^2=7 x^6+42 x^5+5 x^4+85 x^3+10 x^2+79 x+56$
• $y^2=71 x^6+34 x^5+73 x^4+63 x^3+73 x^2+34 x+71$
• $y^2=71 x^6+36 x^5+88 x^4+8 x^3+88 x^2+36 x+71$
• $y^2=84 x^6+40 x^5+17 x^4+44 x^3+17 x^2+40 x+84$
• $y^2=8 x^6+69 x^5+86 x^4+13 x^3+86 x^2+69 x+8$
• $y^2=48 x^6+46 x^5+32 x^4+58 x^3+32 x^2+46 x+48$
• $y^2=7 x^6+75 x^5+49 x^4+43 x^3+71 x^2+43 x+86$
• $y^2=20 x^6+56 x^5+49 x^4+86 x^3+64 x^2+33 x+32$
• $y^2=80 x^6+60 x^5+80 x^4+71 x^3+57 x^2+63 x+2$
• $y^2=2 x^6+68 x^5+61 x^4+74 x^3+37 x^2+42 x+18$
• $y^2=83 x^6+21 x^5+42 x^4+81 x^3+42 x^2+21 x+83$
• $y^2=5 x^6+79 x^5+43 x^4+77 x^3+43 x^2+79 x+5$
• $y^2=52 x^6+37 x^5+33 x^4+57 x^3+66 x^2+59 x+60$
• $y^2=12 x^6+5 x^5+63 x^4+85 x^3+58 x^2+47 x+14$

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.q 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.q : \(\Q(\sqrt{-1}) \).

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.abe_pm	$2$	(not in LMFDB)
2.89.ac_abu	$2$	(not in LMFDB)
2.89.c_abu	$2$	(not in LMFDB)