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

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

Invariants

Base field:	$\F_{73}$
Dimension:	$2$
L-polynomial:	$1 - 5 x + 148 x^{2} - 365 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.413985897192$, $\pm0.491831850955$
Angle rank:	$2$ (numerical)
Number field:	4.0.5717576.2
Galois group:	$D_{4}$
Jacobians:	$66$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5108$	$29871584$	$151724109056$	$806016706514816$	$4297417439363237268$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$69$	$5601$	$390018$	$28382625$	$2072971069$	$151334955822$	$11047406015253$	$806460067935489$	$58871586281020914$	$4297625829961614961$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 66 curves (of which all are hyperelliptic):

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

• $y^2=55 x^6+22 x^5+4 x^4+37 x^3+38 x^2+67 x+69$
• $y^2=22 x^6+49 x^5+65 x^4+66 x^3+52 x^2+16 x+49$
• $y^2=24 x^6+20 x^5+x^4+44 x^3+52 x^2+71 x+41$
• $y^2=20 x^6+66 x^5+42 x^4+20 x^3+22 x^2+52 x+63$
• $y^2=16 x^6+21 x^5+13 x^4+51 x^3+10 x^2+62 x+4$
• $y^2=9 x^6+20 x^5+7 x^4+16 x^3+21 x^2+51 x+3$
• $y^2=5 x^6+24 x^5+7 x^4+5 x^3+38 x^2+16 x+3$
• $y^2=9 x^6+66 x^5+5 x^4+71 x^3+45 x^2+x+7$
• $y^2=55 x^6+16 x^5+65 x^4+51 x^3+38 x^2+62 x+40$
• $y^2=14 x^6+28 x^5+66 x^4+71 x^3+43 x^2+11 x+33$
• $y^2=60 x^6+48 x^5+47 x^4+40 x^3+64 x^2+24 x+26$
• $y^2=61 x^6+10 x^5+45 x^4+11 x^3+49 x^2+54 x+60$
• $y^2=34 x^6+17 x^5+32 x^4+63 x^3+62 x^2+17 x+14$
• $y^2=63 x^6+42 x^5+13 x^4+41 x^3+20 x^2+66 x+28$
• $y^2=25 x^6+70 x^5+68 x^4+11 x^3+33 x^2+10 x+20$
• $y^2=71 x^6+31 x^5+37 x^3+54 x^2+33 x+29$
• $y^2=58 x^6+22 x^5+5 x^4+35 x^3+6 x^2+18 x+59$
• $y^2=17 x^6+9 x^5+41 x^4+47 x^3+47 x^2+54 x+34$
• $y^2=37 x^6+41 x^5+53 x^4+66 x^3+64 x^2+3 x+62$
• $y^2=70 x^6+x^5+45 x^4+47 x^3+22 x^2+35 x+34$
• $y^2=53 x^6+66 x^5+59 x^4+39 x^3+6 x^2+15 x+48$
• $y^2=55 x^6+40 x^4+2 x^3+68 x^2+23 x+25$
• $y^2=10 x^6+40 x^5+70 x^4+11 x^3+6 x^2+59 x+9$
• $y^2=63 x^6+31 x^4+37 x^3+70 x^2+69 x+15$
• $y^2=24 x^6+69 x^5+24 x^4+38 x^3+44 x^2+33 x+13$
• $y^2=49 x^6+55 x^5+37 x^4+25 x^3+72 x^2+29 x+58$
• $y^2=28 x^6+3 x^5+52 x^4+18 x^3+50 x^2+68 x+70$
• $y^2=64 x^6+25 x^5+13 x^4+52 x^3+20 x^2+55 x+17$
• $y^2=18 x^6+47 x^5+72 x^4+64 x^3+59 x^2+60 x+67$
• $y^2=56 x^6+59 x^5+36 x^4+25 x^3+39 x^2+43 x+59$
• $y^2=52 x^6+61 x^5+14 x^4+7 x^3+40 x^2+3 x+1$
• $y^2=68 x^6+33 x^5+28 x^4+49 x^3+12 x^2+67 x+19$
• $y^2=68 x^6+33 x^5+57 x^4+22 x^3+44 x^2+12 x+46$
• $y^2=53 x^6+25 x^5+4 x^4+61 x^2+12 x+36$
• $y^2=8 x^6+63 x^5+46 x^4+58 x^3+33 x^2+52 x+59$
• $y^2=41 x^6+32 x^5+63 x^4+56 x^3+49 x^2+60 x+13$
• $y^2=52 x^6+15 x^5+58 x^4+28 x^3+10 x^2+35 x+61$
• $y^2=61 x^6+39 x^5+9 x^4+43 x^3+57 x^2+22 x+50$
• $y^2=29 x^6+39 x^5+19 x^4+12 x^3+17 x^2+24 x+70$
• $y^2=66 x^6+5 x^5+62 x^4+51 x^3+33 x^2+17 x+60$
• $y^2=47 x^6+61 x^5+2 x^4+29 x^3+2 x^2+55 x+52$
• $y^2=62 x^6+39 x^5+39 x^4+26 x^3+41 x^2+32 x+45$
• $y^2=31 x^5+69 x^4+61 x^3+67 x^2+9$
• $y^2=40 x^6+29 x^5+58 x^4+27 x^3+72 x^2+42 x+7$
• $y^2=29 x^6+48 x^5+25 x^4+70 x^3+9 x^2+63 x+19$
• $y^2=72 x^6+33 x^5+43 x^4+64 x^3+54 x^2+32 x+67$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The endomorphism algebra of this simple isogeny class is 4.0.5717576.2.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.73.f_fs	$2$	(not in LMFDB)