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

• Label: 2.73.ak_gp
• 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:	$( 1 - 5 x + 73 x^{2} )^{2}$
	$1 - 10 x + 171 x^{2} - 730 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.405478609088$, $\pm0.405478609088$
Angle rank:	$1$ (numerical)
Jacobians:	$24$
Cyclic group of points:	no
Non-cyclic primes:	$3, 23$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4761$	$29713401$	$152090640144$	$806233944159081$	$4297249683239995161$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$5572$	$390958$	$28390276$	$2072890144$	$151333900558$	$11047410136288$	$806460173758468$	$58871586269534974$	$4297625821533792772$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-267}) \)$)$

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.a_er	$2$	(not in LMFDB)
2.73.k_gp	$2$	(not in LMFDB)
2.73.f_abw	$3$	(not in LMFDB)