Abelian variety isogeny class 2.73.ao_hn over $\F_{73}$

• Label: 2.73.ao_hn
• Base field: $\F_{73}$
• 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:	$\F_{73}$
Dimension:	$2$
L-polynomial:	$( 1 - 7 x + 73 x^{2} )^{2}$
	$1 - 14 x + 195 x^{2} - 1022 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.365652920247$, $\pm0.365652920247$
Angle rank:	$1$ (numerical)
Jacobians:	$43$
Cyclic group of points:	no
Non-cyclic primes:	$67$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4489$	$29452329$	$152262283264$	$806531089059081$	$4297301914224680089$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$5524$	$391398$	$28400740$	$2072915340$	$151332950158$	$11047400992716$	$806460202367044$	$58871587301004534$	$4297625825788187764$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 43 curves (of which all are hyperelliptic):

• $y^2=59 x^6+5 x^5+59 x^4+45 x^3+56 x^2+7 x+20$
• $y^2=47 x^6+57 x^5+72 x^4+44 x^3+29 x^2+65 x+30$
• $y^2=53 x^6+42 x^5+20 x^4+17 x^3+57 x^2+32 x+50$
• $y^2=29 x^6+24 x^5+28 x^4+46 x^3+71 x^2+20 x+61$
• $y^2=5 x^6+63 x^3+33$
• $y^2=65 x^6+60 x^5+28 x^4+36 x^3+8 x^2+41 x+11$
• $y^2=5 x^6+30 x^3+58$
• $y^2=69 x^6+61 x^5+x^3+23 x+35$
• $y^2=63 x^6+70 x^5+35 x^4+13 x^3+57 x^2+48 x+52$
• $y^2=52 x^6+42 x^5+55 x^4+20 x^3+7 x^2+50 x+27$
• $y^2=5 x^6+5 x^3+45$
• $y^2=21 x^6+12 x^5+56 x^4+19 x^3+38 x^2+16 x+18$
• $y^2=42 x^6+70 x^5+69 x^3+66 x^2+10 x+50$
• $y^2=10 x^6+67 x^5+52 x^4+8 x^3+26 x^2+35 x+56$
• $y^2=18 x^6+67 x^5+66 x^4+30 x^3+61 x^2+11 x+49$
• $y^2=27 x^6+19 x^5+47 x^4+40 x^3+43 x^2+47 x+9$
• $y^2=26 x^6+36 x^5+49 x^4+66 x^3+63 x^2+67 x+1$
• $y^2=14 x^6+57 x^5+45 x^4+5 x^3+17 x^2+30 x+60$
• $y^2=5 x^6+66 x^3+33$
• $y^2=28 x^6+51 x^5+68 x^4+50 x^3+31 x^2+42 x+58$
• and

• $y^2=5 x^6+61 x^5+72 x^4+32 x^3+69 x^2+27 x+28$
• $y^2=39 x^6+3 x^5+70 x^4+64 x^3+34 x^2+64 x+32$
• $y^2=20 x^6+44 x^5+29 x^4+53 x^3+44 x^2+44 x+53$
• $y^2=19 x^6+14 x^5+65 x^4+54 x^3+6 x^2+20 x+44$
• $y^2=68 x^6+56 x^5+31 x^4+60 x^3+31 x^2+56 x+68$
• $y^2=70 x^6+35 x^5+58 x^4+48 x^3+58 x^2+35 x+70$
• $y^2=25 x^6+17 x^5+x^4+67 x^3+27 x^2+56 x+55$
• $y^2=61 x^6+7 x^5+72 x^4+69 x^3+8 x^2+5 x+71$
• $y^2=68 x^6+22 x^5+72 x^4+50 x^2+31 x+47$
• $y^2=63 x^6+50 x^5+35 x^4+10 x^3+48 x^2+27 x+56$
• $y^2=24 x^6+24 x^5+2 x^4+16 x^3+12 x^2+61 x+1$
• $y^2=17 x^6+6 x^5+45 x^4+13 x^3+63 x^2+3 x+63$
• $y^2=41 x^6+3 x^5+29 x^4+2 x^3+55 x^2+71 x+50$
• $y^2=59 x^6+21 x^5+45 x^4+3 x^3+59 x^2+60 x+53$
• $y^2=21 x^6+37 x^5+56 x^4+4 x^3+33 x^2+x+43$
• $y^2=25 x^6+49 x^5+13 x^4+59 x^3+14 x^2+24 x+18$
• $y^2=13 x^6+41 x^5+64 x^4+7 x^3+69 x^2+9 x+44$
• $y^2=33 x^6+61 x^5+70 x^4+68 x^3+x^2+23 x+15$
• $y^2=39 x^6+61 x^5+60 x^4+5 x^3+14 x^2+12 x+42$
• $y^2=11 x^6+17 x^5+11 x^4+45 x^3+x^2+70 x+4$
• $y^2=5 x^6+65 x^5+9 x^4+27 x^3+x^2+72 x+68$
• $y^2=40 x^6+32 x^5+12 x^4+53 x^3+57 x^2+65 x+62$
• $y^2=69 x^6+31 x^5+46 x^4+12 x^3+40 x^2+3 x+30$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73}$.

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.ah^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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
2.73.a_dt	$2$	(not in LMFDB)
2.73.o_hn	$2$	(not in LMFDB)
2.73.au_jm	$3$	(not in LMFDB)
2.73.ar_ii	$3$	(not in LMFDB)
2.73.h_ay	$3$	(not in LMFDB)
2.73.k_bb	$3$	(not in LMFDB)
2.73.bi_qt	$3$	(not in LMFDB)