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

• Label: 2.73.ac_fr
• 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:	$( 1 - x + 73 x^{2} )^{2}$
	$1 - 2 x + 147 x^{2} - 146 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.481361674224$, $\pm0.481361674224$
Angle rank:	$1$ (numerical)
Jacobians:	$56$
Cyclic group of points:	no
Non-cyclic primes:	$73$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5329$	$29975625$	$151504663696$	$805871447015625$	$4297516869751267969$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$5620$	$389454$	$28377508$	$2073019032$	$151335687310$	$11047403817144$	$806459990537668$	$58871586220154142$	$4297625836614462100$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 56 curves (of which all are hyperelliptic):

• $y^2=18 x^6+64 x^5+24 x^4+46 x^3+3 x^2+x+18$
• $y^2=12 x^6+11 x^5+6 x^4+48 x^3+13 x^2+38 x+30$
• $y^2=59 x^6+33 x^5+14 x^4+57 x^3+20 x^2+45 x+59$
• $y^2=15 x^6+26 x^5+17 x^4+71 x^3+49 x^2+31 x+9$
• $y^2=28 x^6+64 x^5+50 x^4+31 x^3+70 x^2+3 x+24$
• $y^2=25 x^6+59 x^5+46 x^4+59 x^3+x^2+14 x+55$
• $y^2=2 x^6+61 x^5+42 x^4+35 x^3+71 x^2+34 x+46$
• $y^2=24 x^6+71 x^5+7 x^4+48 x^3+17 x^2+15 x+14$
• $y^2=32 x^6+53 x^5+29 x^4+22 x^3+69 x^2+4 x+51$
• $y^2=x^6+63 x^5+31 x^4+72 x^3+31 x^2+31$
• $y^2=2 x^6+12 x^5+20 x^4+38 x^3+56 x^2+33 x+29$
• $y^2=x^6+28 x^5+17 x^4+26 x^3+43 x^2+68 x+27$
• $y^2=66 x^6+64 x^5+67 x^4+43 x^3+53 x^2+21 x+20$
• $y^2=64 x^6+31 x^5+14 x^4+42 x^3+10 x^2+62 x+8$
• $y^2=45 x^6+67 x^5+11 x^4+28 x^3+51 x^2+49 x+5$
• $y^2=49 x^6+12 x^5+56 x^4+47 x^3+42 x^2+62 x+43$
• $y^2=48 x^6+52 x^5+53 x^4+9 x^3+2 x^2+63 x+13$
• $y^2=36 x^6+6 x^5+22 x^4+45 x^3+7 x^2+54 x+50$
• $y^2=60 x^6+32 x^5+53 x^4+29 x^3+3 x^2+60 x+35$
• $y^2=61 x^6+9 x^5+7 x^4+44 x^3+10 x^2+72 x+61$
• and

• $y^2=22 x^6+62 x^5+42 x^4+34 x^3+31 x^2+62 x+51$
• $y^2=13 x^6+41 x^5+19 x^4+52 x^3+44 x^2+10 x+42$
• $y^2=27 x^6+28 x^5+21 x^4+56 x^3+67 x^2+69 x+16$
• $y^2=18 x^6+39 x^5+18 x^4+51 x^3+71 x^2+14 x+18$
• $y^2=27 x^6+20 x^5+53 x^4+68 x^3+33 x^2+7 x+70$
• $y^2=29 x^6+9 x^5+45 x^4+35 x^3+15 x^2+x+20$
• $y^2=67 x^6+62 x^5+72 x^4+56 x^3+39 x^2+2 x+60$
• $y^2=51 x^6+33 x^5+45 x^4+x^3+68 x^2+38 x+37$
• $y^2=23 x^6+37 x^5+53 x^4+27 x^3+53 x^2+35 x+34$
• $y^2=41 x^6+51 x^5+59 x^4+60 x^3+34 x^2+47 x+23$
• $y^2=63 x^6+55 x^5+30 x^4+35 x^3+42 x^2+64 x+66$
• $y^2=51 x^6+65 x^5+42 x^4+62 x^3+14 x^2+64 x+10$
• $y^2=26 x^6+26 x^5+54 x^4+70 x^3+17 x^2+13 x+16$
• $y^2=24 x^6+69 x^5+6 x^4+4 x^3+x^2+8 x+65$
• $y^2=41 x^6+19 x^5+64 x^4+20 x^3+45 x^2+28 x+38$
• $y^2=51 x^6+11 x^5+57 x^4+33 x^3+24 x^2+43 x+56$
• $y^2=66 x^6+46 x^5+5 x^4+5 x^3+3 x^2+60 x+11$
• $y^2=31 x^6+4 x^5+43 x^4+35 x^3+30 x^2+4 x+42$
• $y^2=63 x^6+71 x^5+3 x^4+15 x^3+10 x^2+45 x+4$
• $y^2=54 x^6+47 x^5+6 x^4+64 x^3+24 x^2+40 x+71$
• $y^2=44 x^6+14 x^5+47 x^4+21 x^3+18 x^2+46 x+40$
• $y^2=7 x^6+40 x^5+8 x^4+56 x^3+6 x^2+72 x+13$
• $y^2=16 x^6+36 x^5+22 x^4+60 x^3+55 x^2+x+41$
• $y^2=17 x^6+8 x^5+15 x^4+40 x^3+21 x^2+44 x+66$
• $y^2=52 x^6+22 x^5+19 x^4+10 x^3+x^2+24 x+32$
• $y^2=48 x^6+39 x^5+60 x^4+22 x^3+18 x^2+36 x+42$
• $y^2=14 x^6+69 x^5+7 x^4+39 x^3+28 x^2+9 x+20$
• $y^2=21 x^6+15 x^5+70 x^4+23 x^3+25 x^2+65 x+25$
• $y^2=5 x^6+17 x^3+33$
• $y^2=5 x^6+14 x^3+28$
• $y^2=70 x^6+15 x^5+46 x^4+6 x^3+41 x^2+43 x+72$
• $y^2=38 x^6+56 x^5+3 x^4+2 x^3+35 x^2+14 x+50$
• $y^2=10 x^6+22 x^5+16 x^4+54 x^3+49 x^2+7 x+64$
• $y^2=26 x^6+29 x^5+44 x^4+59 x^3+32 x^2+62 x+23$
• $y^2=3 x^6+58 x^5+7 x^4+63 x^3+15 x^2+22 x+58$
• $y^2=25 x^6+48 x^5+30 x^4+41 x^3+52 x^2+62 x+46$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-291}) \)$)$

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_fp	$2$	(not in LMFDB)
2.73.c_fr	$2$	(not in LMFDB)
2.73.b_acu	$3$	(not in LMFDB)