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

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

Invariants

Base field:	$\F_{73}$
Dimension:	$2$
L-polynomial:	$( 1 + 8 x + 73 x^{2} )^{2}$
	$1 + 16 x + 210 x^{2} + 1168 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.655084565757$, $\pm0.655084565757$
Angle rank:	$1$ (numerical)
Jacobians:	$42$
Cyclic group of points:	no
Non-cyclic primes:	$2, 41$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6724$	$29289744$	$150371777284$	$806683601534976$	$4297870658799332164$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$5494$	$386538$	$28406110$	$2073189690$	$151332707158$	$11047402051146$	$806460174534334$	$58871585789306394$	$4297625831022511414$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

• $y^2=68 x^6+29 x^5+22 x^4+38 x^3+17 x^2+5 x+27$
• $y^2=2 x^6+30 x^5+32 x^4+68 x^3+32 x^2+30 x+2$
• $y^2=65 x^6+67 x^5+32 x^4+38 x^3+67 x^2+27 x+13$
• $y^2=12 x^6+68 x^5+49 x^4+46 x^3+49 x^2+68 x+12$
• $y^2=41 x^6+67 x^5+40 x^4+43 x^3+20 x^2+70 x+40$
• $y^2=3 x^6+45 x^5+36 x^4+55 x^3+25 x^2+59 x+46$
• $y^2=70 x^6+38 x^5+60 x^4+60 x^3+36 x^2+69 x+50$
• $y^2=23 x^6+18 x^5+42 x^4+40 x^3+28 x^2+36 x+23$
• $y^2=36 x^6+34 x^5+46 x^4+33 x^3+46 x^2+34 x+36$
• $y^2=48 x^6+60 x^4+60 x^2+48$
• $y^2=43 x^6+57 x^5+9 x^4+10 x^3+50 x^2+37 x+63$
• $y^2=12 x^6+11 x^5+41 x^4+25 x^3+31 x^2+64 x+16$
• $y^2=46 x^6+30 x^5+41 x^4+30 x^3+48 x+50$
• $y^2=60 x^6+70 x^5+70 x^4+22 x^3+27 x^2+72 x+61$
• $y^2=x^6+48 x^5+18 x^3+67 x^2+35 x+4$
• $y^2=24 x^6+71 x^5+9 x^4+4 x^3+11 x^2+38 x+63$
• $y^2=x^6+11 x^5+45 x^4+70 x^3+44 x^2+6 x+33$
• $y^2=71 x^6+15 x^5+7 x^4+31 x^3+7 x^2+33 x+46$
• $y^2=54 x^6+66 x^5+44 x^4+27 x^3+37 x^2+12 x+25$
• $y^2=65 x^6+7 x^5+69 x^4+63 x^3+41 x^2+37 x+10$
• and

• $y^2=37 x^6+42 x^5+69 x^4+43 x^3+54 x^2+26 x+61$
• $y^2=3 x^6+39 x^5+71 x^4+36 x^3+26 x^2+47 x+46$
• $y^2=72 x^6+49 x^5+3 x^4+62 x^3+49 x^2+27 x+50$
• $y^2=2 x^6+20 x^5+48 x^4+58 x^3+11 x^2+45 x+12$
• $y^2=14 x^6+66 x^5+28 x^4+35 x^3+25 x^2+45 x+65$
• $y^2=36 x^6+19 x^5+52 x^4+72 x^3+39 x^2+70 x+38$
• $y^2=36 x^6+40 x^5+66 x^4+42 x^3+66 x^2+40 x+36$
• $y^2=13 x^6+36 x^5+31 x^4+54 x^3+66 x^2+60 x+39$
• $y^2=64 x^6+3 x^5+26 x^4+5 x^3+2 x^2+28 x+54$
• $y^2=40 x^6+25 x^5+12 x^4+45 x^3+12 x^2+25 x+40$
• $y^2=5 x^6+7 x^5+14 x^4+48 x^3+48 x^2+50 x+51$
• $y^2=68 x^6+33 x^4+33 x^2+68$
• $y^2=41 x^6+30 x^5+70 x^4+44 x^3+19 x^2+58 x+39$
• $y^2=45 x^6+31 x^5+66 x^4+24 x^3+5 x^2+62 x+40$
• $y^2=22 x^6+59 x^5+65 x^4+5 x^3+58 x^2+33 x+62$
• $y^2=18 x^6+56 x^5+38 x^4+20 x^3+29 x^2+28 x+71$
• $y^2=39 x^6+49 x^5+59 x^4+4 x^3+18 x^2+58 x+72$
• $y^2=40 x^6+30 x^5+44 x^4+20 x^3+23 x^2+45 x+6$
• $y^2=5 x^6+10 x^3+40$
• $y^2=57 x^6+66 x^5+26 x^4+40 x^3+47 x^2+31 x+42$
• $y^2=32 x^6+36 x^5+52 x^4+19 x^3+52 x^2+36 x+32$
• $y^2=5 x^6+44 x^3+26$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.i^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-57}) \)$)$

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.aq_ic	$2$	(not in LMFDB)
2.73.a_de	$2$	(not in LMFDB)
2.73.ai_aj	$3$	(not in LMFDB)