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

• Label: 2.73.ad_acm
• Base field: $\F_{73}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{73}$
Dimension:	$2$
L-polynomial:	$1 - 3 x - 64 x^{2} - 219 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.110492508693$, $\pm0.777159175360$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-283})\)
Galois group:	$C_2^2$
Jacobians:	$63$
Isomorphism classes:	99
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5044$	$27681472$	$150845238544$	$806690467425024$	$4297480047962105524$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$71$	$5193$	$387758$	$28406353$	$2073001271$	$151334988558$	$11047404796079$	$806460100885921$	$58871587678658174$	$4297625830502738793$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 63 curves (of which all are hyperelliptic):

• $y^2=72 x^6+59 x^5+63 x^4+9 x^3+67 x^2+69 x+48$
• $y^2=64 x^6+29 x^5+11 x^4+56 x^3+26 x^2+13 x+24$
• $y^2=8 x^6+29 x^5+5 x^4+3 x^3+49 x^2+61 x+45$
• $y^2=50 x^6+63 x^5+55 x^4+53 x^3+71 x^2+66 x+64$
• $y^2=30 x^6+59 x^5+37 x^4+5 x^3+27 x^2+38 x+70$
• $y^2=30 x^6+2 x^5+46 x^4+5 x^3+51 x^2+14 x+34$
• $y^2=14 x^6+37 x^5+44 x^4+15 x^3+22 x^2+2 x+33$
• $y^2=68 x^5+68 x^4+68 x^3+23 x^2+31 x+15$
• $y^2=50 x^6+72 x^5+5 x^4+32 x^3+20 x^2+50 x+35$
• $y^2=10 x^6+28 x^5+52 x^4+59 x^3+67 x^2+43 x+28$
• $y^2=56 x^6+25 x^5+47 x^4+26 x^3+20 x^2+62 x+6$
• $y^2=15 x^6+28 x^5+39 x^4+59 x^3+26 x^2+72 x+52$
• $y^2=5 x^6+20 x^5+18 x^4+6 x^3+34 x^2+50 x+55$
• $y^2=68 x^6+36 x^5+13 x^3+30 x^2+19 x+64$
• $y^2=13 x^6+25 x^5+43 x^4+70 x^3+2 x^2+7$
• $y^2=70 x^6+24 x^5+29 x^4+19 x^3+53 x^2+63 x+35$
• $y^2=52 x^6+45 x^5+46 x^4+58 x^3+6 x^2+72 x+51$
• $y^2=17 x^6+35 x^5+18 x^4+10 x^3+52 x^2+67 x+16$
• $y^2=3 x^6+67 x^5+6 x^4+45 x^2+45 x+65$
• $y^2=34 x^6+51 x^5+34 x^4+56 x^3+45 x^2+25 x+30$
• and

• $y^2=35 x^6+27 x^5+34 x^4+21 x^3+20 x^2+42 x$
• $y^2=21 x^6+3 x^5+70 x^4+35 x^3+46 x^2+59 x+61$
• $y^2=54 x^6+53 x^5+66 x^4+14 x^3+68 x^2+15 x+11$
• $y^2=17 x^6+15 x^5+50 x^4+19 x^3+x^2+32 x+7$
• $y^2=6 x^6+43 x^5+45 x^4+40 x^3+51 x^2+59 x+58$
• $y^2=38 x^6+9 x^5+10 x^4+16 x^3+34 x^2+71 x+8$
• $y^2=30 x^6+42 x^5+13 x^4+47 x^3+34 x^2+53 x+12$
• $y^2=27 x^6+48 x^5+26 x^4+7 x^3+18 x^2+30 x+20$
• $y^2=30 x^6+19 x^5+27 x^4+10 x^3+27 x^2+46 x+55$
• $y^2=39 x^6+30 x^5+15 x^4+11 x^3+71 x^2+45 x+8$
• $y^2=70 x^6+8 x^5+x^4+46 x^3+43 x^2+8 x$
• $y^2=14 x^6+15 x^5+58 x^4+48 x^2+71 x+29$
• $y^2=57 x^6+32 x^5+29 x^4+50 x^3+29 x^2+65 x+29$
• $y^2=45 x^6+54 x^5+58 x^4+26 x^3+24 x^2+12 x+22$
• $y^2=x^6+18 x^5+45 x^4+60 x^3+44 x^2+63 x+51$
• $y^2=31 x^6+25 x^5+44 x^4+64 x^3+11 x^2+23 x+17$
• $y^2=23 x^6+63 x^5+47 x^4+25 x^3+23 x^2+13 x+43$
• $y^2=18 x^6+20 x^5+27 x^4+16 x^3+32 x^2+24 x+63$
• $y^2=62 x^6+72 x^5+41 x^4+19 x^3+11 x^2+58 x+63$
• $y^2=24 x^6+54 x^5+49 x^4+18 x^3+12 x^2+12 x+36$
• $y^2=56 x^6+66 x^5+61 x^4+19 x^3+14 x^2+4 x+16$
• $y^2=44 x^6+49 x^5+10 x^4+30 x^3+38 x^2+x+66$
• $y^2=36 x^6+40 x^5+35 x^4+47 x^3+69 x^2+37 x+10$
• $y^2=19 x^6+24 x^5+35 x^4+24 x^3+3 x^2+62 x+69$
• $y^2=23 x^6+23 x^5+53 x^4+64 x^3+63 x^2+57 x+31$
• $y^2=58 x^6+67 x^5+43 x^4+42 x^3+8 x^2+20 x+43$
• $y^2=32 x^6+9 x^5+8 x^4+6 x^3+70 x^2+59 x+47$
• $y^2=66 x^6+50 x^5+60 x^4+66 x^3+12 x^2+61 x+31$
• $y^2=43 x^6+32 x^5+52 x^4+28 x^3+63 x^2+10 x+56$
• $y^2=21 x^6+54 x^5+43 x^4+16 x^3+61 x^2+22 x+66$
• $y^2=34 x^6+58 x^5+18 x^4+64 x^3+59 x^2+34 x+31$
• $y^2=9 x^6+x^5+56 x^4+49 x^3+27 x^2+34 x+66$
• $y^2=66 x^6+6 x^5+x^4+40 x^3+17 x^2+50 x+27$
• $y^2=28 x^6+23 x^5+x^4+46 x^3+54 x^2+24 x+46$
• $y^2=72 x^6+48 x^5+20 x^4+39 x^3+9 x^2+48 x+49$
• $y^2=27 x^6+40 x^5+14 x^4+57 x^3+70 x^2+9 x+44$
• $y^2=49 x^6+36 x^5+63 x^4+59 x^3+62 x^2+15 x+13$
• $y^2=19 x^6+31 x^5+36 x^4+33 x^3+33 x^2+63 x+42$
• $y^2=54 x^6+13 x^5+39 x^4+15 x^3+39 x^2+59 x+22$
• $y^2=3 x^5+29 x^4+52 x^3+48 x^2+34 x+23$
• $y^2=51 x^6+68 x^5+56 x^4+48 x^3+9 x^2+41 x+45$
• $y^2=14 x^6+7 x^5+16 x^4+36 x^3+43 x^2+34 x+11$
• $y^2=63 x^6+23 x^5+32 x^4+37 x^3+8 x^2+2 x+30$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73^{3}}$.

Endomorphism algebra over $\F_{73}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-283})\).

Endomorphism algebra over $\overline{\F}_{73}$

The base change of $A$ to $\F_{73^{3}}$ is 1.389017.ayg^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.d_acm	$2$	(not in LMFDB)
2.73.g_fz	$3$	(not in LMFDB)
2.73.ag_fz	$6$	(not in LMFDB)