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

• Label: 2.73.ah_ay
• 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:	$( 1 - 17 x + 73 x^{2} )( 1 + 10 x + 73 x^{2} )$
	$1 - 7 x - 24 x^{2} - 511 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.0323195869136$, $\pm0.698986253580$
Angle rank:	$1$ (numerical)
Jacobians:	$33$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4788$	$27885312$	$150410557584$	$806424595651584$	$4297463870869889748$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$67$	$5233$	$386638$	$28396993$	$2072993467$	$151332950158$	$11047399755907$	$806460036657601$	$58871586115531294$	$4297625831661242593$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 33 curves (of which all are hyperelliptic):

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

• $y^2=24 x^6+42 x^5+60 x^4+x^3+21 x^2+39 x+55$
• $y^2=6 x^6+11 x^5+31 x^4+62 x^3+66 x^2+3 x+14$
• $y^2=44 x^6+9 x^5+58 x^4+11 x^3+57 x^2+7 x+5$
• $y^2=32 x^6+43 x^5+43 x^4+66 x^3+62 x^2+60 x+4$
• $y^2=42 x^6+20 x^5+37 x^4+35 x^3+25 x^2+70 x+56$
• $y^2=38 x^6+38 x^5+54 x^4+55 x^3+59 x^2+69 x+34$
• $y^2=51 x^6+66 x^5+6 x^4+46 x^3+38 x^2+46 x+35$
• $y^2=58 x^6+28 x^5+66 x^4+31 x^3+69 x^2+25 x+22$
• $y^2=60 x^6+55 x^5+50 x^4+39 x^3+52 x^2+32 x+36$
• $y^2=54 x^6+72 x^5+22 x^4+70 x^3+43 x^2+18 x+35$
• $y^2=54 x^6+50 x^5+10 x^4+47 x^3+60 x^2+53 x+37$
• $y^2=15 x^6+68 x^5+23 x^4+19 x^3+37 x^2+43 x+40$
• $y^2=46 x^6+58 x^5+13 x^4+45 x^3+28 x^2+59 x+30$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.ar $\times$ 1.73.k 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.ar : \(\Q(\sqrt{-3}) \).
• 1.73.k : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{73^{3}}$ is 1.389017.abtu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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.abb_me	$2$	(not in LMFDB)
2.73.h_ay	$2$	(not in LMFDB)
2.73.bb_me	$2$	(not in LMFDB)
2.73.abi_qt	$3$	(not in LMFDB)
2.73.ak_bb	$3$	(not in LMFDB)
2.73.o_hn	$3$	(not in LMFDB)
2.73.r_ii	$3$	(not in LMFDB)
2.73.u_jm	$3$	(not in LMFDB)