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

• Label: 2.73.aj_i
• 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:	$1 - 9 x + 8 x^{2} - 657 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.00989864991722$, $\pm0.656768016749$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-211})\)
Galois group:	$C_2^2$
Jacobians:	$15$
Isomorphism classes:	15
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4672$	$28050688$	$150370226176$	$806277418988544$	$4297557896221229632$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$65$	$5265$	$386534$	$28391809$	$2073038825$	$151332697230$	$11047394030465$	$806460076481089$	$58871585775493622$	$4297625826631221825$

Jacobians and polarizations

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

• $y^2=12 x^6+x^5+13 x^4+68 x^3+19 x^2+39 x+33$
• $y^2=51 x^6+56 x^5+7 x^4+49 x^3+22 x^2+44 x+69$
• $y^2=20 x^6+49 x^5+37 x^4+49 x^3+31 x^2+53 x+63$
• $y^2=34 x^6+6 x^5+67 x^4+41 x^3+56 x^2+25 x+2$
• $y^2=3 x^6+44 x^5+10 x^4+9 x^3+23 x^2+25 x+37$
• $y^2=58 x^6+57 x^5+33 x^4+16 x^3+x^2+17 x+46$
• $y^2=26 x^6+14 x^5+22 x^4+8 x^3+41 x^2+5 x+29$
• $y^2=45 x^6+7 x^5+60 x^4+50 x^3+72 x^2+16 x+16$
• $y^2=62 x^6+54 x^5+12 x^4+38 x^3+34 x^2+52 x+53$
• $y^2=41 x^6+9 x^5+71 x^4+64 x^3+4 x^2+65 x+6$
• $y^2=55 x^6+12 x^5+30 x^4+34 x^3+70 x^2+59 x+19$
• $y^2=51 x^6+37 x^5+10 x^4+44 x^3+9 x^2+42 x+63$
• $y^2=39 x^6+25 x^5+39 x^4+21 x^3+22 x^2+5 x+13$
• $y^2=27 x^6+64 x^5+2 x^4+8 x^3+16 x^2+23 x+2$
• $y^2=40 x^6+13 x^5+70 x^4+52 x^3+72 x^2+23 x$

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 \(\Q(\sqrt{-3}, \sqrt{-211})\).

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

The base change of $A$ to $\F_{73^{3}}$ is 1.389017.abvu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-211}) \)$)$

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.j_i	$2$	(not in LMFDB)
2.73.s_it	$3$	(not in LMFDB)
2.73.as_it	$6$	(not in LMFDB)