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

• Label: 2.89.abe_pn
• 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:	$( 1 - 15 x + 89 x^{2} )^{2}$
	$1 - 30 x + 403 x^{2} - 2670 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.207471636293$, $\pm0.207471636293$
Angle rank:	$1$ (numerical)
Jacobians:	$40$
Cyclic group of points:	no
Non-cyclic primes:	$3, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5625$	$62015625$	$497871360000$	$3938299847015625$	$31183377591212015625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$7828$	$706230$	$62769508$	$5584356300$	$496983317038$	$44231338867020$	$3936588684953668$	$350356401542796390$	$31181719908242460148$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=23 x^6+60 x^5+31 x^4+16 x^3+70 x^2+13 x+28$
• $y^2=55 x^6+37 x^5+82 x^4+18 x^3+86 x^2+74 x+9$
• $y^2=85 x^6+77 x^5+67 x^4+59 x^3+78 x^2+86 x+44$
• $y^2=44 x^6+71 x^5+52 x^4+61 x^3+38 x^2+47 x+2$
• $y^2=46 x^6+28 x^5+67 x^4+33 x^3+67 x^2+28 x+46$
• $y^2=37 x^6+25 x^5+54 x^4+53 x^3+76 x^2+87 x+76$
• $y^2=17 x^6+82 x^5+81 x^4+20 x^3+47 x^2+24 x+50$
• $y^2=60 x^6+70 x^5+3 x^4+75 x^3+6 x^2+13 x+35$
• $y^2=43 x^6+64 x^5+47 x^4+40 x^3+88 x^2+85 x+30$
• $y^2=79 x^6+79 x^5+28 x^4+66 x^3+86 x^2+21 x+20$
• $y^2=34 x^6+8 x^5+17 x^4+15 x^3+13 x^2+15 x+32$
• $y^2=24 x^6+64 x^5+78 x^4+4 x^3+59 x^2+34 x+80$
• $y^2=24 x^6+85 x^5+55 x^4+74 x^3+67 x^2+7 x+1$
• $y^2=82 x^6+20 x^5+55 x^4+54 x^3+57 x^2+56 x+63$
• $y^2=82 x^6+38 x^5+48 x^4+84 x^3+48 x^2+38 x+82$
• $y^2=74 x^6+40 x^5+78 x^4+69 x^3+34 x^2+21 x+83$
• $y^2=31 x^6+49 x^5+47 x^4+56 x^3+80 x^2+69 x+51$
• $y^2=18 x^6+57 x^5+69 x^4+76 x^3+72 x^2+73 x+21$
• $y^2=29 x^6+17 x^5+72 x^4+71 x^3+5 x^2+80 x+31$
• $y^2=52 x^6+43 x^5+79 x^4+66 x^3+22 x^2+23 x+75$
• and

• $y^2=70 x^6+33 x^5+42 x^4+17 x^3+11 x^2+23 x+82$
• $y^2=78 x^6+25 x^5+22 x^4+88 x^3+5 x^2+32 x+68$
• $y^2=3 x^6+21 x^5+2 x^4+34 x^3+55 x^2+17 x+35$
• $y^2=79 x^6+76 x^5+27 x^4+39 x^3+58 x^2+19 x+21$
• $y^2=33 x^6+69 x^5+4 x^4+9 x^3+25 x^2+42 x+23$
• $y^2=49 x^6+15 x^5+80 x^4+88 x^3+9 x^2+15 x+40$
• $y^2=5 x^6+36 x^5+65 x^4+50 x^3+74 x^2+53 x+8$
• $y^2=36 x^6+71 x^5+40 x^4+46 x^3+55 x^2+80 x+47$
• $y^2=23 x^6+48 x^5+53 x^4+66 x^3+79 x^2+7 x+82$
• $y^2=87 x^6+2 x^5+76 x^4+19 x^3+75 x^2+85 x+18$
• $y^2=19 x^6+82 x^5+16 x^4+55 x^3+73 x^2+82 x+70$
• $y^2=73 x^6+7 x^5+19 x^4+58 x^3+61 x^2+83 x+36$
• $y^2=67 x^6+63 x^5+44 x^4+28 x^3+23 x^2+44 x+31$
• $y^2=55 x^6+16 x^5+22 x^4+23 x^3+4 x^2+2 x+68$
• $y^2=88 x^6+45 x^5+60 x^4+28 x^3+23 x^2+50 x+21$
• $y^2=79 x^6+66 x^5+74 x^4+47 x^3+82 x^2+12 x+45$
• $y^2=83 x^6+46 x^5+13 x^4+17 x^3+61 x^2+77 x+13$
• $y^2=51 x^6+52 x^5+35 x^4+70 x^3+13 x^2+35 x+35$
• $y^2=58 x^6+23 x^5+87 x^4+42 x^3+40 x^2+33 x+46$
• $y^2=76 x^6+50 x^5+73 x^4+86 x^3+34 x^2+45 x+46$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.ap^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-131}) \)$)$

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_abv	$2$	(not in LMFDB)
2.89.be_pn	$2$	(not in LMFDB)
2.89.p_fg	$3$	(not in LMFDB)