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

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

Invariants

Base field:	$\F_{73}$
Dimension:	$2$
L-polynomial:	$1 + 16 x + 208 x^{2} + 1168 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.625937907550$, $\pm0.685725039985$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-226 +16 \sqrt{2}})\)
Galois group:	$D_{4}$
Jacobians:	$36$
Isomorphism classes:	36
Cyclic group of points:	yes

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6722$	$29267588$	$150409005026$	$806672909244944$	$4297841136466074882$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$90$	$5490$	$386634$	$28405734$	$2073175450$	$151332938130$	$11047401481290$	$806460151207614$	$58871586094237722$	$4297625830595590450$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

• $y^2=50 x^6+23 x^5+3 x^4+58 x^3+54 x^2+61 x+40$
• $y^2=48 x^6+32 x^5+3 x^4+16 x^3+58 x^2+18 x+68$
• $y^2=49 x^6+72 x^5+41 x^4+31 x^3+45 x^2+34 x+16$
• $y^2=4 x^6+9 x^5+59 x^4+72 x^3+7 x^2+38 x+46$
• $y^2=34 x^6+17 x^5+31 x^3+63 x^2+15 x+32$
• $y^2=11 x^6+12 x^5+2 x^4+55 x^3+41 x^2+19 x+43$
• $y^2=28 x^6+40 x^5+20 x^4+28 x^3+x^2+10 x+22$
• $y^2=55 x^6+44 x^5+62 x^4+37 x^3+71 x^2+40 x+58$
• $y^2=42 x^6+52 x^5+49 x^4+60 x^3+x^2+22 x+48$
• $y^2=8 x^6+13 x^5+28 x^4+55 x^3+6 x^2+55 x+5$
• $y^2=29 x^6+4 x^5+65 x^4+55 x^3+x^2+71 x+41$
• $y^2=61 x^6+40 x^5+13 x^4+9 x^3+44 x^2+12 x+40$
• $y^2=14 x^6+33 x^5+5 x^4+17 x^3+22 x^2+47 x+71$
• $y^2=43 x^6+43 x^5+44 x^4+70 x^3+55 x^2+51 x+18$
• $y^2=6 x^6+33 x^5+62 x^4+9 x^3+37 x^2+59 x+16$
• $y^2=55 x^6+42 x^5+17 x^4+34 x^3+5 x^2+67 x+61$
• $y^2=61 x^6+7 x^5+2 x^4+53 x^3+59 x^2+42 x+56$
• $y^2=35 x^6+44 x^5+42 x^4+30 x^3+71 x^2+34 x+24$
• $y^2=70 x^6+6 x^5+61 x^4+32 x^3+14 x^2+70 x+40$
• $y^2=57 x^6+22 x^5+57 x^4+37 x^3+48 x^2+23 x+39$
• and

• $y^2=46 x^6+27 x^5+4 x^4+15 x^3+61 x^2+25 x+45$
• $y^2=9 x^6+26 x^5+14 x^4+28 x^3+11 x^2+36 x+36$
• $y^2=19 x^6+7 x^5+45 x^4+71 x^3+47 x^2+54 x+42$
• $y^2=21 x^6+2 x^5+2 x^4+69 x^3+20 x^2+19 x+32$
• $y^2=43 x^6+67 x^5+69 x^4+17 x^3+16 x^2+12 x+13$
• $y^2=10 x^6+9 x^5+57 x^4+35 x^3+63 x^2+8 x+62$
• $y^2=37 x^6+50 x^5+54 x^4+44 x^3+9 x^2+35 x+48$
• $y^2=3 x^6+12 x^5+45 x^4+38 x^3+66 x^2+69 x+55$
• $y^2=28 x^6+47 x^5+67 x^4+45 x^3+18 x^2+36 x+38$
• $y^2=4 x^6+50 x^5+51 x^4+42 x^3+18 x^2+13 x+57$
• $y^2=15 x^6+61 x^5+8 x^4+23 x^3+36 x^2+46 x+46$
• $y^2=32 x^6+62 x^5+52 x^4+14 x^3+3 x^2+53 x+49$
• $y^2=46 x^6+24 x^5+45 x^4+39 x^3+4 x^2+63 x+63$
• $y^2=19 x^6+27 x^5+44 x^4+13 x^3+60 x^2+37 x+64$
• $y^2=62 x^6+62 x^5+58 x^4+66 x^3+34 x^2+46 x+18$
• $y^2=46 x^6+19 x^5+67 x^4+65 x^3+x^2+64 x+37$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-226 +16 \sqrt{2}})\).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.73.aq_ia	$2$	(not in LMFDB)