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

• Label: 2.73.ag_fz
• 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:	$( 1 - 3 x + 73 x^{2} )^{2}$
	$1 - 6 x + 155 x^{2} - 438 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.443825842026$, $\pm0.443825842026$
Angle rank:	$1$ (numerical)
Jacobians:	$39$
Cyclic group of points:	no
Non-cyclic primes:	$71$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5041$	$29888089$	$151825563904$	$805999538197161$	$4297334269567615441$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$68$	$5604$	$390278$	$28382020$	$2072930948$	$151334988558$	$11047411073060$	$806460073910404$	$58871585737877654$	$4297625828105195364$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 39 curves (of which all are hyperelliptic):

• $y^2=16 x^6+38 x^5+57 x^4+33 x^3+64 x^2+24 x+71$
• $y^2=59 x^6+9 x^5+60 x^4+53 x^3+50 x^2+23 x+58$
• $y^2=40 x^6+72 x^5+45 x^4+31 x^3+39 x^2+66 x+4$
• $y^2=17 x^6+20 x^5+29 x^4+41 x^3+40 x^2+66 x+22$
• $y^2=23 x^6+71 x^5+63 x^4+19 x^3+39 x^2+2 x+58$
• $y^2=34 x^6+23 x^5+4 x^4+34 x^3+14 x^2+19 x+7$
• $y^2=66 x^6+46 x^5+16 x^4+67 x^3+24 x^2+13 x+26$
• $y^2=42 x^6+64 x^5+32 x^4+30 x^3+26 x^2+20 x+67$
• $y^2=71 x^6+32 x^5+2 x^4+21 x^3+9 x^2+53 x+59$
• $y^2=26 x^6+71 x^5+20 x^4+55 x^3+41 x^2+48 x+42$
• $y^2=34 x^6+56 x^5+63 x^4+63 x^3+4 x^2+42 x+1$
• $y^2=34 x^6+43 x^5+7 x^4+52 x^3+72 x^2+49 x+48$
• $y^2=10 x^6+22 x^5+46 x^4+20 x^3+36 x^2+41 x+54$
• $y^2=50 x^6+x^5+23 x^4+50 x^3+54 x^2+38 x+56$
• $y^2=46 x^6+21 x^5+7 x^4+56 x^3+53 x^2+47 x+13$
• $y^2=65 x^6+69 x^5+10 x^4+55 x^3+23 x^2+28 x+20$
• $y^2=19 x^6+18 x^5+62 x^4+43 x^3+23 x^2+44 x+35$
• $y^2=38 x^6+7 x^5+34 x^4+44 x^3+44 x^2+63 x+69$
• $y^2=26 x^6+38 x^5+30 x^4+43 x^3+44 x^2+48 x+62$
• $y^2=38 x^6+31 x^5+60 x^4+15 x^3+53 x^2+44 x+4$
• and

• $y^2=31 x^6+8 x^5+11 x^4+68 x^3+23 x^2+28 x+52$
• $y^2=46 x^6+72 x^5+25 x^4+10 x^3+61 x^2+41 x+3$
• $y^2=38 x^6+50 x^5+34 x^4+26 x^3+22 x^2+3 x+36$
• $y^2=35 x^6+39 x^5+28 x^4+16 x^3+8 x^2+x+31$
• $y^2=69 x^6+44 x^5+17 x^4+15 x^3+63 x^2+42 x+69$
• $y^2=62 x^6+23 x^5+44 x^4+65 x^3+38 x^2+47 x+60$
• $y^2=24 x^6+21 x^5+30 x^4+20 x^3+60 x^2+48 x+2$
• $y^2=17 x^6+6 x^5+50 x^4+67 x^2+35 x+30$
• $y^2=x^6+59 x^5+65 x^4+5 x^3+32 x^2+68 x+9$
• $y^2=29 x^6+10 x^5+15 x^4+29 x^3+44 x^2+29 x+33$
• $y^2=31 x^6+13 x^5+28 x^4+70 x^3+67 x^2+41 x+65$
• $y^2=47 x^6+23 x^5+53 x^4+42 x^2+7 x+20$
• $y^2=11 x^6+5 x^5+62 x^4+58 x^3+22 x^2+20 x+58$
• $y^2=16 x^6+56 x^5+22 x^4+34 x^3+61 x^2+57 x+4$
• $y^2=34 x^6+2 x^5+58 x^4+4 x^3+21 x^2+x+20$
• $y^2=61 x^6+6 x^5+24 x^4+46 x^3+48 x^2+24 x+50$
• $y^2=38 x^6+58 x^5+8 x^4+51 x^3+24 x^2+68 x+14$
• $y^2=4 x^6+27 x^5+54 x^4+54 x^3+31 x^2+42 x+9$
• $y^2=49 x^6+8 x^5+42 x^4+11 x^3+30 x^2+44 x+50$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.ad^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-283}) \)$)$

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_fh	$2$	(not in LMFDB)
2.73.g_fz	$2$	(not in LMFDB)
2.73.d_acm	$3$	(not in LMFDB)