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

• Label: 2.73.a_afq
• Base field: $\F_{73}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{73}$
Dimension:	$2$
L-polynomial:	$( 1 - 73 x^{2} )^{2}$
	$1 - 146 x^{2} + 5329 x^{4}$
Frobenius angles:	$0$, $0$, $1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{73}) \)
Galois group:	$C_2$
Jacobians:	$13$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5184$	$26873856$	$151333448256$	$805854925357056$	$4297625825557414464$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$5038$	$389018$	$28376926$	$2073071594$	$151332670222$	$11047398519098$	$806459978301118$	$58871586708267914$	$4297625821411271278$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 13 curves (of which all are hyperelliptic):

• $y^2=10 x^6+7 x^4+25 x^3+64 x^2+46$
• $y^2=x^6+26 x^3+56$
• $y^2=43 x^6+51 x^5+64 x^4+6 x^3+43 x^2+7 x+65$
• $y^2=x^6+x^3+66$
• $y^2=49 x^6+70 x^5+15 x^4+19 x^3+68 x^2+10 x+53$
• $y^2=54 x^6+49 x^5+10 x^4+52 x^2+27 x+2$
• $y^2=52 x^6+45 x^5+65 x^4+32 x^3+x^2+4 x+67$
• $y^2=41 x^6+6 x^5+33 x^4+14 x^3+5 x^2+20 x+43$
• $y^2=x^6+11 x^3+17$
• $y^2=x^6+x^3+17$
• $y^2=69 x^6+55 x^5+34 x^4+42 x^3+32 x^2+3 x+45$
• $y^2=47 x^6+62 x^5+30 x^4+43 x^3+57 x^2+53 x+23$
• $y^2=31 x^6+22 x^5+68 x^4+65 x^3+2 x^2+59 x+57$

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 the quaternion algebra over \(\Q(\sqrt{73}) \) ramified at both real infinite places.

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

The base change of $A$ to $\F_{73^{2}}$ is 1.5329.afq^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $73$ and $\infty$.

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_cv	$3$	(not in LMFDB)
2.73.a_fq	$4$	(not in LMFDB)
2.73.a_cv	$6$	(not in LMFDB)