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

• Label: 2.89.i_hm
• 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:	$( 1 + 4 x + 89 x^{2} )^{2}$
	$1 + 8 x + 194 x^{2} + 712 x^{3} + 7921 x^{4}$
Frobenius angles:	$\pm0.567997546099$, $\pm0.567997546099$
Angle rank:	$1$ (numerical)
Jacobians:	$60$
Cyclic group of points:	no
Non-cyclic primes:	$2, 47$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8836$	$65351056$	$495568129156$	$3935283749785600$	$31183182579809491396$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$98$	$8246$	$702962$	$62721438$	$5584321378$	$496982094806$	$44231308368562$	$3936588840267838$	$350356405930122338$	$31181719917999282806$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 60 curves (of which all are hyperelliptic):

• $y^2=72 x^6+48 x^5+25 x^4+66 x^3+55 x^2+65 x+81$
• $y^2=33 x^5+20 x^4+88 x^2+7 x+74$
• $y^2=65 x^6+67 x^5+20 x^4+75 x^3+13 x^2+18 x+4$
• $y^2=76 x^6+71 x^5+73 x^4+61 x^3+88 x^2+80 x+13$
• $y^2=8 x^6+15 x^5+68 x^4+62 x^3+45 x^2+28 x+55$
• $y^2=52 x^6+83 x^5+45 x^4+48 x^3+64 x^2+19 x+10$
• $y^2=x^6+39 x^5+76 x^4+10 x^3+59 x^2+78 x+49$
• $y^2=41 x^6+74 x^5+72 x^4+66 x^3+66 x^2+21 x+26$
• $y^2=x^6+77 x^5+57 x^4+41 x^3+16 x^2+57 x+45$
• $y^2=40 x^6+20 x^5+79 x^4+8 x^3+17 x^2+40 x+42$
• $y^2=46 x^6+8 x^5+24 x^4+41 x^3+47 x^2+36 x+2$
• $y^2=86 x^6+17 x^5+31 x^4+42 x^3+31 x^2+17 x+86$
• $y^2=57 x^6+22 x^5+78 x^4+88 x^3+30 x^2+6 x+87$
• $y^2=15 x^6+10 x^5+18 x^4+88 x^3+29 x^2+53 x+79$
• $y^2=11 x^6+45 x^5+23 x^4+74 x^3+14 x^2+44 x+78$
• $y^2=40 x^6+72 x^5+86 x^4+71 x^3+15 x^2+18 x+60$
• $y^2=28 x^6+4 x^5+49 x^4+24 x^3+49 x^2+4 x+28$
• $y^2=36 x^6+75 x^5+20 x^4+54 x^3+20 x^2+75 x+36$
• $y^2=29 x^6+76 x^5+84 x^4+18 x^3+8 x^2+7 x+16$
• $y^2=35 x^6+4 x^5+39 x^4+52 x^3+25 x^2+23 x+18$
• and

• $y^2=84 x^6+57 x^5+61 x^4+69 x^3+81 x^2+39 x+51$
• $y^2=47 x^6+16 x^5+77 x^4+80 x^3+51 x^2+60 x+34$
• $y^2=3 x^5+76 x^4+27 x^3+81 x^2+78 x+36$
• $y^2=42 x^6+47 x^5+55 x^4+70 x^3+60 x^2+75 x+75$
• $y^2=46 x^6+64 x^5+63 x^4+36 x^3+7 x^2+23 x+29$
• $y^2=44 x^6+55 x^4+55 x^2+44$
• $y^2=65 x^6+83 x^5+39 x^4+17 x^3+x^2+39 x+6$
• $y^2=72 x^6+53 x^5+5 x^4+83 x^3+24 x^2+14 x+38$
• $y^2=71 x^6+55 x^5+50 x^4+40 x^3+44 x^2+49 x+40$
• $y^2=72 x^6+18 x^5+4 x^4+60 x^3+52 x^2+21 x+82$
• $y^2=54 x^6+57 x^5+32 x^4+49 x^3+45 x^2+55 x+84$
• $y^2=49 x^6+28 x^5+80 x^4+46 x^3+58 x^2+14 x+53$
• $y^2=19 x^6+78 x^5+85 x^4+78 x^3+30 x^2+84 x+68$
• $y^2=58 x^6+76 x^5+37 x^4+49 x^3+4 x^2+66 x+67$
• $y^2=66 x^6+57 x^5+53 x^4+2 x^3+35 x^2+24 x+47$
• $y^2=17 x^6+84 x^5+28 x^4+59 x^3+13 x+71$
• $y^2=74 x^6+48 x^5+41 x^4+82 x^3+54 x^2+56 x+31$
• $y^2=84 x^6+50 x^5+19 x^4+75 x^3+19 x^2+50 x+84$
• $y^2=48 x^6+20 x^5+6 x^4+41 x^3+6 x^2+20 x+48$
• $y^2=64 x^6+49 x^5+62 x^4+66 x^3+41 x^2+45 x+55$
• $y^2=65 x^6+39 x^4+39 x^2+65$
• $y^2=69 x^6+32 x^5+x^4+22 x^3+43 x^2+81 x+36$
• $y^2=28 x^6+6 x^5+13 x^4+73 x^3+59 x^2+23 x+33$
• $y^2=68 x^6+33 x^5+52 x^4+35 x^3+45 x^2+63 x+70$
• $y^2=2 x^6+11 x^5+23 x^4+74 x^3+23 x^2+11 x+2$
• $y^2=36 x^6+68 x^5+77 x^4+84 x^3+41 x^2+52 x+81$
• $y^2=76 x^6+68 x^5+69 x^4+18 x^3+83 x^2+72 x+58$
• $y^2=86 x^6+58 x^5+22 x^4+68 x^3+36 x^2+48 x+87$
• $y^2=61 x^6+73 x^5+21 x^4+33 x^3+88 x^2+16 x+25$
• $y^2=85 x^6+63 x^5+32 x^4+39 x^3+29 x^2+66 x+3$
• $y^2=68 x^6+26 x^5+12 x^4+19 x^3+69 x^2+88 x+40$
• $y^2=79 x^6+33 x^5+50 x^4+64 x^3+44 x^2+86 x+71$
• $y^2=10 x^6+7 x^5+40 x^4+66 x^3+86 x^2+35 x+88$
• $y^2=66 x^6+56 x^5+22 x^4+25 x^3+14 x^2+63 x+58$
• $y^2=58 x^6+51 x^5+81 x^4+21 x^3+81 x^2+51 x+58$
• $y^2=46 x^6+42 x^5+49 x^4+5 x^3+39 x^2+8 x+54$
• $y^2=87 x^5+52 x^4+61 x^3+36 x^2+81 x+57$
• $y^2=67 x^6+88 x^5+37 x^4+37 x^3+37 x^2+88 x+67$
• $y^2=87 x^6+65 x^5+45 x^4+57 x^3+64 x^2+68 x+84$
• $y^2=74 x^6+28 x^5+21 x^4+7 x^3+71 x^2+46 x+72$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{89}$

The isogeny class factors as 1.89.e^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-85}) \)$)$

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.ai_hm	$2$	(not in LMFDB)
2.89.a_gg	$2$	(not in LMFDB)
2.89.ae_acv	$3$	(not in LMFDB)