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

• Label: 2.73.a_afk
• 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:	$1 - 140 x^{2} + 5329 x^{4}$
Frobenius angles:	$\pm0.0457860281612$, $\pm0.954213971839$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-6}, \sqrt{286})\)
Galois group:	$C_2^2$
Jacobians:	$8$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5190$	$26936100$	$151333720470$	$805952354490000$	$4297625829156268950$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$5050$	$389018$	$28380358$	$2073071594$	$151333214650$	$11047398519098$	$806460045568318$	$58871586708267914$	$4297625828608980250$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=23 x^6+3 x^5+49 x^4+24 x^3+46 x^2+31 x+11$
• $y^2=42 x^6+15 x^5+26 x^4+47 x^3+11 x^2+9 x+55$
• $y^2=67 x^6+x^5+10 x^4+9 x^3+19 x^2+1$
• $y^2=43 x^6+5 x^5+50 x^4+45 x^3+22 x^2+5$
• $y^2=31 x^6+2 x^5+61 x^4+19 x^3+71 x^2+13 x+14$
• $y^2=9 x^6+10 x^5+13 x^4+22 x^3+63 x^2+65 x+70$
• $y^2=42 x^6+17 x^5+37 x^4+36 x^3+67 x^2+42 x+35$
• $y^2=64 x^6+12 x^5+39 x^4+34 x^3+43 x^2+64 x+29$

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 \(\Q(\sqrt{-6}, \sqrt{286})\).

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

The base change of $A$ to $\F_{73^{2}}$ is 1.5329.afk^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-429}) \)$)$

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.a_fk	$4$	(not in LMFDB)