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

• Label: 2.73.ax_km
• 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:	$( 1 - 14 x + 73 x^{2} )( 1 - 9 x + 73 x^{2} )$
	$1 - 23 x + 272 x^{2} - 1679 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.194368965322$, $\pm0.323434683416$
Angle rank:	$2$ (numerical)
Jacobians:	$36$
Isomorphism classes:	234
Cyclic group of points:	no
Non-cyclic primes:	$2, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3900$	$28485600$	$151943828400$	$806874559920000$	$4297745938426297500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$51$	$5345$	$390582$	$28412833$	$2073129531$	$151334136110$	$11047396849827$	$806460097550593$	$58871586832250886$	$4297625828694458225$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

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

• $y^2=50 x^6+51 x^5+33 x^4+49 x^3+30 x^2+29 x+63$
• $y^2=16 x^6+70 x^5+47 x^4+44 x^3+17 x^2+65 x+29$
• $y^2=40 x^6+56 x^5+32 x^4+56 x^3+56 x^2+66 x+8$
• $y^2=51 x^6+58 x^5+22 x^4+47 x^3+46 x^2+69 x+19$
• $y^2=38 x^6+16 x^5+17 x^4+63 x^3+29 x^2+x+21$
• $y^2=9 x^6+13 x^5+64 x^4+56 x^3+72 x+63$
• $y^2=22 x^6+53 x^5+2 x^3+29 x^2+20 x+32$
• $y^2=34 x^6+43 x^5+12 x^4+35 x^3+13 x^2+46 x+47$
• $y^2=59 x^6+32 x^5+43 x^4+38 x^3+52 x^2+16 x+56$
• $y^2=2 x^6+63 x^5+36 x^4+55 x^3+30 x^2+7 x+43$
• $y^2=31 x^6+52 x^5+31 x^4+34 x^3+32 x^2+13 x+64$
• $y^2=72 x^6+30 x^5+40 x^4+38 x^3+54 x^2+11 x+70$
• $y^2=56 x^6+21 x^5+42 x^4+46 x^3+22 x^2+35 x+45$
• $y^2=53 x^6+38 x^5+6 x^4+35 x^3+49 x^2+42 x+18$
• $y^2=27 x^6+56 x^5+7 x^4+25 x^3+19 x^2+35 x+33$
• $y^2=66 x^6+59 x^5+10 x^4+10 x^3+26 x^2+29 x+54$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.ao $\times$ 1.73.aj and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.73.ao : \(\Q(\sqrt{-6}) \).
• 1.73.aj : \(\Q(\sqrt{-211}) \).

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.af_u	$2$	(not in LMFDB)
2.73.f_u	$2$	(not in LMFDB)
2.73.x_km	$2$	(not in LMFDB)