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

• Label: 2.73.d_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:	$\pm0.222840824640$, $\pm0.889507491307$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-283})\)
Galois group:	$C_2^2$
Jacobians:	$63$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5488$	$27681472$	$151825563904$	$806690467425024$	$4297771617189515248$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$77$	$5193$	$390278$	$28406353$	$2073141917$	$151334988558$	$11047392242117$	$806460100885921$	$58871585737877654$	$4297625830502738793$

Jacobians and polarizations

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

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

• $y^2=7 x^6+20 x^5+36 x^4+48 x^3+4 x^2+23 x$
• $y^2=32 x^6+15 x^5+58 x^4+29 x^3+11 x^2+3 x+13$
• $y^2=51 x^6+46 x^5+38 x^4+70 x^3+48 x^2+2 x+55$
• $y^2=12 x^6+2 x^5+31 x^4+22 x^3+5 x^2+14 x+35$
• $y^2=30 x^6+69 x^5+6 x^4+54 x^3+36 x^2+3 x+71$
• $y^2=44 x^6+45 x^5+50 x^4+7 x^3+24 x^2+63 x+40$
• $y^2=4 x^6+64 x^5+65 x^4+16 x^3+24 x^2+46 x+60$
• $y^2=62 x^6+21 x^5+57 x^4+35 x^3+17 x^2+4 x+27$
• $y^2=4 x^6+22 x^5+62 x^4+50 x^3+62 x^2+11 x+56$
• $y^2=49 x^6+4 x^5+2 x^4+55 x^3+63 x^2+6 x+40$
• $y^2=58 x^6+40 x^5+5 x^4+11 x^3+69 x^2+40 x$
• $y^2=32 x^6+3 x^5+70 x^4+68 x^2+58 x+35$
• $y^2=66 x^6+14 x^5+72 x^4+31 x^3+72 x^2+33 x+72$
• $y^2=9 x^6+40 x^5+70 x^4+49 x^3+34 x^2+17 x+19$
• $y^2=5 x^6+17 x^5+6 x^4+8 x^3+x^2+23 x+36$
• $y^2=50 x^6+5 x^5+38 x^4+42 x^3+46 x^2+63 x+18$
• $y^2=42 x^6+23 x^5+16 x^4+52 x^3+42 x^2+65 x+69$
• $y^2=62 x^6+4 x^5+20 x^4+47 x^3+21 x^2+34 x+71$
• $y^2=18 x^6+68 x^5+59 x^4+22 x^3+55 x^2+71 x+23$
• $y^2=47 x^6+51 x^5+26 x^4+17 x^3+60 x^2+60 x+34$
• $y^2=55 x^6+57 x^5+56 x^4+33 x^3+32 x^2+30 x+47$
• $y^2=38 x^6+39 x^5+2 x^4+6 x^3+66 x^2+44 x+57$
• $y^2=51 x^6+8 x^5+7 x^4+24 x^3+43 x^2+22 x+2$
• $y^2=22 x^6+47 x^5+29 x^4+47 x^3+15 x^2+18 x+53$
• $y^2=63 x^6+63 x^5+69 x^4+42 x^3+71 x^2+26 x+50$
• $y^2=70 x^6+28 x^5+67 x^4+23 x^3+60 x^2+4 x+67$
• $y^2=14 x^6+45 x^5+40 x^4+30 x^3+58 x^2+3 x+16$
• $y^2=57 x^6+10 x^5+12 x^4+57 x^3+17 x^2+56 x+50$
• $y^2=67 x^6+21 x^5+25 x^4+64 x^3+71 x^2+2 x+55$
• $y^2=32 x^6+51 x^5+69 x^4+7 x^3+13 x^2+37 x+38$
• $y^2=24 x^6+71 x^5+17 x^4+28 x^3+3 x^2+24 x+9$
• $y^2=45 x^6+5 x^5+61 x^4+26 x^3+62 x^2+24 x+38$
• $y^2=57 x^6+45 x^5+44 x^4+8 x^3+18 x^2+10 x+20$
• $y^2=64 x^6+63 x^5+44 x^4+53 x^3+40 x^2+34 x+53$
• $y^2=68 x^6+21 x^5+27 x^4+49 x^3+45 x^2+21 x+26$
• $y^2=62 x^6+54 x^5+70 x^4+66 x^3+58 x^2+45 x+1$
• $y^2=26 x^6+34 x^5+23 x^4+3 x^3+18 x^2+2 x+65$
• $y^2=33 x^6+50 x^5+51 x^4+65 x^3+65 x^2+71 x+23$
• $y^2=51 x^6+65 x^5+49 x^4+2 x^3+49 x^2+3 x+37$
• $y^2=59 x^5+35 x^4+25 x^3+68 x^2+36 x+63$
• $y^2=36 x^6+48 x^5+61 x^4+21 x^3+45 x^2+59 x+6$
• $y^2=32 x^6+16 x^5+47 x^4+51 x^3+67 x^2+36 x+46$
• $y^2=23 x^6+42 x^5+14 x^4+39 x^3+40 x^2+10 x+4$

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.yg^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.ad_acm	$2$	(not in LMFDB)
2.73.ag_fz	$3$	(not in LMFDB)
2.73.ad_acm	$6$	(not in LMFDB)