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

• Label: 2.73.ac_acr
• 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:	$1 - 2 x - 69 x^{2} - 146 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.129325725892$, $\pm0.795992392559$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
Galois group:	$C_2^2$
Jacobians:	$15$
Isomorphism classes:	238
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5113$	$27656217$	$151000633744$	$806730107548329$	$4297521343216366873$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$5188$	$388158$	$28407748$	$2073021192$	$151335412558$	$11047403384712$	$806460125461636$	$58871587552917774$	$4297625828097776068$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):

• $y^2=58 x^6+58 x^5+13 x^4+48 x^3+4 x^2+5 x+28$
• $y^2=45 x^6+14 x^5+64 x^4+50 x^3+6 x^2+29 x+49$
• $y^2=5 x^6+5 x^3+42$
• $y^2=5 x^6+23 x^5+28 x^4+51 x^3+30 x^2+52 x+48$
• $y^2=5 x^6+25 x^3+63$
• $y^2=5 x^6+5 x^3+59$
• $y^2=5 x^6+5 x^3+31$
• $y^2=72 x^6+25 x^5+50 x^4+48 x^3+32 x^2+49 x+19$
• $y^2=64 x^6+17 x^5+35 x^4+11 x^3+61 x^2+65 x+7$
• $y^2=5 x^6+25 x^3+10$
• $y^2=x^6+3 x^5+13 x^4+22 x^3+43 x^2+64 x+1$
• $y^2=5 x^6+5 x^3+60$
• $y^2=5 x^6+25 x^3+66$
• $y^2=39 x^6+45 x^5+65 x^4+40 x^3+9 x^2+72 x+45$
• $y^2=5 x^6+5 x^3+20$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73^{3}}$.

Endomorphism algebra over $\F_{73}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-3})\).

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

The base change of $A$ to $\F_{73^{3}}$ is 1.389017.aqo^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.c_acr	$2$	(not in LMFDB)
2.73.e_fu	$3$	(not in LMFDB)
2.73.ae_fu	$6$	(not in LMFDB)