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

• Label: 2.73.c_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:	$\pm0.204007607441$, $\pm0.870674274108$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
Galois group:	$C_2^2$
Jacobians:	$15$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5409$	$27656217$	$151669744704$	$806730107548329$	$4297730317125328449$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$76$	$5188$	$389878$	$28407748$	$2073121996$	$151335412558$	$11047393653484$	$806460125461636$	$58871585863618054$	$4297625828097776068$

Jacobians and polarizations

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

• $y^2=71 x^6+71 x^5+65 x^4+21 x^3+20 x^2+25 x+67$
• $y^2=9 x^6+32 x^5+42 x^4+10 x^3+45 x^2+35 x+39$
• $y^2=x^6+x^3+23$
• $y^2=25 x^6+42 x^5+67 x^4+36 x^3+4 x^2+41 x+21$
• $y^2=x^6+5 x^3+71$
• $y^2=x^6+x^3+41$
• $y^2=x^6+x^3+50$
• $y^2=68 x^6+52 x^5+31 x^4+21 x^3+14 x^2+26 x+22$
• $y^2=42 x^6+18 x^5+7 x^4+46 x^3+56 x^2+13 x+16$
• $y^2=x^6+5 x^3+2$
• $y^2=5 x^6+15 x^5+65 x^4+37 x^3+69 x^2+28 x+5$
• $y^2=x^6+x^3+12$
• $y^2=x^6+5 x^3+57$
• $y^2=49 x^6+6 x^5+33 x^4+54 x^3+45 x^2+68 x+6$
• $y^2=x^6+x^3+4$

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.qo^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.ac_acr	$2$	(not in LMFDB)
2.73.ae_fu	$3$	(not in LMFDB)
2.73.a_fm	$6$	(not in LMFDB)