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

• Label: 2.73.as_gg
• 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:	$1 - 18 x + 162 x^{2} - 1314 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.0174767392761$, $\pm0.482523260724$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{65})\)
Galois group:	$C_2^2$
Jacobians:	$32$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4160$	$28387840$	$150935620160$	$805869459865600$	$4297392951296744000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$5330$	$387992$	$28377438$	$2072959256$	$151334226290$	$11047394848952$	$806459989083838$	$58871586050896376$	$4297625829703557650$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 32 curves (of which all are hyperelliptic):

• $y^2=38 x^6+58 x^5+29 x^4+55 x^3+53 x^2+44 x+48$
• $y^2=32 x^6+10 x^5+26 x^4+26 x^2+63 x+32$
• $y^2=4 x^6+29 x^5+32 x^4+11 x^3+9 x^2+62 x+14$
• $y^2=23 x^6+33 x^5+51 x^4+46 x^3+20 x^2+62 x+45$
• $y^2=25 x^6+64 x^5+71 x^4+45 x^3+35 x^2+61 x+39$
• $y^2=26 x^6+65 x^5+62 x^4+62 x^3+x^2+52 x+58$
• $y^2=28 x^6+56 x^5+8 x^4+14 x^3+7 x^2+69 x+54$
• $y^2=28 x^6+44 x^5+68 x^4+65 x^3+61 x^2+47 x+7$
• $y^2=68 x^6+44 x^5+3 x^4+55 x^3+20 x^2+7 x+55$
• $y^2=11 x^6+5 x^5+48 x^4+15 x^3+39 x^2+59 x+3$
• $y^2=66 x^6+16 x^5+4 x^4+22 x^3+63 x^2+6 x+72$
• $y^2=43 x^6+52 x^5+19 x^4+67 x^3+27 x^2+55 x+14$
• $y^2=22 x^6+9 x^5+3 x^4+3 x^2+64 x+22$
• $y^2=68 x^6+60 x^5+33 x^4+33 x^2+13 x+68$
• $y^2=13 x^6+20 x^5+64 x^3+52 x^2+35 x+64$
• $y^2=57 x^6+56 x^5+63 x^4+38 x^3+62 x^2+63 x+52$
• $y^2=56 x^6+2 x^5+57 x^4+49 x^3+2 x^2+28 x+10$
• $y^2=31 x^6+55 x^5+31 x^4+31 x^2+18 x+31$
• $y^2=44 x^6+18 x^5+23 x^4+69 x^3+54 x^2+12 x+62$
• $y^2=65 x^6+30 x^5+53 x^4+67 x^3+23 x^2+31 x+26$
• and

• $y^2=60 x^6+5 x^5+58 x^4+63 x^3+6 x^2+50 x+39$
• $y^2=13 x^6+34 x^5+6 x^4+51 x^3+8 x^2+19 x+67$
• $y^2=38 x^6+54 x^5+11 x^4+11 x^3+18 x^2+47 x+15$
• $y^2=42 x^6+59 x^5+70 x^4+34 x^3+36 x^2+72 x+5$
• $y^2=7 x^6+63 x^5+61 x^4+40 x^3+55 x^2+15 x+63$
• $y^2=4 x^6+38 x^5+9 x^4+47 x^3+55 x^2+65 x+37$
• $y^2=18 x^6+26 x^5+62 x^4+29 x^3+18 x^2+42 x+59$
• $y^2=45 x^6+9 x^5+72 x^4+61 x^3+44 x^2+63 x+11$
• $y^2=55 x^6+30 x^5+61 x^4+45 x^3+26 x^2+17 x+10$
• $y^2=44 x^6+17 x^5+60 x^4+51 x^3+15 x^2+41 x+51$
• $y^2=17 x^6+68 x^5+53 x^4+5 x^3+21 x^2+5 x+31$
• $y^2=13 x^6+44 x^5+33 x^4+x^3+25 x^2+67 x+22$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73^{4}}$.

Endomorphism algebra over $\F_{73}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{65})\).

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

The base change of $A$ to $\F_{73^{4}}$ is 1.28398241.apkc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-65}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{73^{2}}$

  The base change of $A$ to $\F_{73^{2}}$ is the simple isogeny class 2.5329.a_apkc and its endomorphism algebra is \(\Q(i, \sqrt{65})\).

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.s_gg	$2$	(not in LMFDB)
2.73.s_gg	$4$	(not in LMFDB)
2.73.a_aq	$8$	(not in LMFDB)