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

• Label: 2.73.a_afe
• 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:	$1 - 134 x^{2} + 5329 x^{4}$
Frobenius angles:	$\pm0.0649785193373$, $\pm0.935021480663$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{70})\)
Galois group:	$C_2^2$
Jacobians:	$20$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5196$	$26998416$	$151333962444$	$806045701211136$	$4297625831583373836$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$5062$	$389018$	$28383646$	$2073071594$	$151333698598$	$11047398519098$	$806460098965438$	$58871586708267914$	$4297625833463190022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):

• $y^2=x^6+42 x^3+66$
• $y^2=x^6+44 x^3+66$
• $y^2=50 x^6+7 x^5+49 x^4+37 x^3+60 x^2+62 x+40$
• $y^2=x^6+29 x^3+7$
• $y^2=64 x^5+59 x^4+52 x^3+37 x^2+12 x$
• $y^2=3 x^6+23 x^5+22 x^4+53 x^3+44 x^2+24 x+23$
• $y^2=15 x^6+42 x^5+37 x^4+46 x^3+x^2+47 x+42$
• $y^2=25 x^6+35 x^5+57 x^4+14 x^3+29 x^2+32 x+42$
• $y^2=69 x^6+18 x^5+71 x^4+25 x^3+29 x^2+25 x+40$
• $y^2=61 x^6+26 x^5+47 x^4+57 x^3+27 x^2+7 x+16$
• $y^2=41 x^6+64 x^5+25 x^4+71 x^3+68 x^2+23 x+68$
• $y^2=15 x^6+60 x^5+13 x^4+35 x^3+57 x^2+14 x+35$
• $y^2=65 x^6+39 x^5+39 x^4+20 x^3+46 x^2+64 x+52$
• $y^2=65 x^6+35 x^5+35 x^4+34 x^3+51 x^2+53 x+46$
• $y^2=x^6+40 x^3+52$
• $y^2=25 x^6+9 x^5+43 x^4+58 x^3+32 x^2+27 x+58$
• $y^2=42 x^6+66 x^5+67 x^4+64 x^3+31 x^2+22 x+25$
• $y^2=36 x^6+8 x^5+18 x^4+71 x^3+57 x^2+46 x+39$
• $y^2=34 x^6+40 x^5+17 x^4+63 x^3+66 x^2+11 x+49$
• $y^2=60 x^6+31 x^5+60 x^4+30 x^3+37 x^2+17 x+25$

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 \(\Q(\sqrt{-3}, \sqrt{70})\).

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

The base change of $A$ to $\F_{73^{2}}$ is 1.5329.afe^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-210}) \)$)$

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.ag_dh	$3$	(not in LMFDB)
2.73.g_dh	$3$	(not in LMFDB)
2.73.a_fe	$4$	(not in LMFDB)