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

• Label: 2.73.a_afm
• 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:	$1 - 142 x^{2} + 5329 x^{4}$
Frobenius angles:	$\pm0.0373409407743$, $\pm0.962659059226$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{8})\)
Galois group:	$C_2^2$
Jacobians:	$14$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5188$	$26915344$	$151333633156$	$805920331677696$	$4297625828097776068$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$5046$	$389018$	$28379230$	$2073071594$	$151333040022$	$11047398519098$	$806460024758974$	$58871586708267914$	$4297625826491994486$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

• $y^2=46 x^6+27 x^5+18 x^4+38 x^2+50 x+65$
• $y^2=28 x^6+12 x^5+13 x^4+59 x^2+12 x+47$
• $y^2=21 x^6+3 x^5+9 x^4+8 x^3+20 x^2+4 x+46$
• $y^2=46 x^6+37 x^5+3 x^4+70 x^2+36 x+27$
• $y^2=37 x^6+35 x^5+15 x^4+21 x^2+19 x+4$
• $y^2=24 x^6+61 x^5+20 x^4+28 x^3+4 x^2+55 x+51$
• $y^2=7 x^6+56 x^5+60 x^4+28 x^3+48 x^2+30 x+3$
• $y^2=66 x^6+60 x^5+63 x^4+21 x^2+42 x+43$
• $y^2=28 x^6+19 x^5+61 x^4+28 x^3+14 x^2+36 x+38$
• $y^2=22 x^6+71 x^5+28 x^4+51 x^2+72 x+66$
• $y^2=62 x^6+7 x^5+64 x^4+37 x^3+43 x^2+21 x+36$
• $y^2=23 x^6+38 x^5+10 x^4+40 x^2+49 x+12$
• $y^2=x^5+48 x$
• $y^2=17 x^6+49 x^5+25 x^4+67 x^3+33 x^2+32 x+46$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73^{2}}$.

Endomorphism algebra over $\F_{73}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\).

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

The base change of $A$ to $\F_{73^{2}}$ is 1.5329.afm^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

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.ae_fu	$4$	(not in LMFDB)
2.73.a_fm	$4$	(not in LMFDB)
2.73.e_fu	$4$	(not in LMFDB)