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

• Label: 2.73.r_ii
• 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:	$( 1 + 7 x + 73 x^{2} )( 1 + 10 x + 73 x^{2} )$
	$1 + 17 x + 216 x^{2} + 1241 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.634347079753$, $\pm0.698986253580$
Angle rank:	$1$ (numerical)
Jacobians:	$45$
Isomorphism classes:	178

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6804$	$29175552$	$150410557584$	$806738206546944$	$4297790802781220724$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$91$	$5473$	$386638$	$28408033$	$2073151171$	$151332950158$	$11047403557099$	$806460130961281$	$58871586115531294$	$4297625831889913393$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 45 curves (of which all are hyperelliptic):

• $y^2=38 x^6+44 x^5+55 x^4+6 x^3+33 x^2+41 x+49$
• $y^2=51 x^6+14 x^5+69 x^4+57 x^3+9 x^2+64 x+49$
• $y^2=42 x^6+26 x^5+29 x^4+63 x^3+72 x^2+3 x+16$
• $y^2=x^6+45 x^4+71 x^3+41 x^2+23 x+51$
• $y^2=51 x^6+22 x^5+58 x^4+45 x^3+35 x^2+23 x+1$
• $y^2=41 x^6+63 x^5+62 x^4+45 x^3+3 x^2+70 x+8$
• $y^2=44 x^6+42 x^5+27 x^4+49 x^3+14 x^2+31 x+18$
• $y^2=x^6+44 x^5+37 x^4+72 x^3+33 x^2+14 x+65$
• $y^2=70 x^6+13 x^5+56 x^4+46 x^3+5 x^2+15 x+63$
• $y^2=24 x^6+5 x^5+65 x^4+67 x^3+60 x^2+34 x+6$
• $y^2=38 x^6+64 x^5+36 x^4+16 x^3+66 x^2+65 x+9$
• $y^2=71 x^6+49 x^5+38 x^4+26 x^3+70 x^2+5 x$
• $y^2=13 x^6+42 x^5+48 x^3+18 x^2+62 x+69$
• $y^2=29 x^6+60 x^5+55 x^4+51 x^3+28 x^2+20 x+13$
• $y^2=72 x^6+67 x^5+33 x^4+15 x^3+13 x^2+49 x+17$
• $y^2=54 x^6+35 x^5+52 x^4+67 x^3+58 x^2+50 x+37$
• $y^2=58 x^6+68 x^5+x^4+67 x^3+38 x^2+51 x+19$
• $y^2=28 x^6+35 x^5+57 x^4+41 x^3+23 x^2+68 x+16$
• $y^2=21 x^6+50 x^5+7 x^4+20 x^3+55 x^2+70 x+69$
• $y^2=x^6+27 x^5+16 x^4+30 x^3+31 x^2+36 x+4$
• and

• $y^2=48 x^6+61 x^5+39 x^4+50 x^3+25 x^2+39 x+2$
• $y^2=61 x^6+59 x^5+12 x^4+18 x^3+20 x^2+67 x+50$
• $y^2=59 x^6+12 x^5+40 x^4+69 x^3+57 x^2+x+55$
• $y^2=57 x^6+47 x^5+3 x^4+36 x^3+x^2+13 x+38$
• $y^2=4 x^6+53 x^5+58 x^4+29 x^3+34 x^2+72 x+4$
• $y^2=52 x^6+72 x^5+61 x^4+49 x^3+43 x^2+24 x+70$
• $y^2=22 x^6+18 x^5+8 x^4+15 x^3+25 x^2+39 x+12$
• $y^2=32 x^6+62 x^5+23 x^4+51 x^3+67 x^2+18 x+17$
• $y^2=11 x^6+60 x^5+66 x^4+23 x^3+39 x^2+30 x+41$
• $y^2=53 x^6+34 x^5+61 x^4+44 x^3+25 x^2+27 x+21$
• $y^2=61 x^6+66 x^5+21 x^4+5 x^3+12 x^2+66 x+16$
• $y^2=9 x^6+69 x^5+43 x^4+6 x^3+64 x^2+38 x+71$
• $y^2=23 x^6+58 x^5+47 x^3+45 x^2+36 x+59$
• $y^2=31 x^6+50 x^5+44 x^4+48 x^3+2 x^2+68 x+50$
• $y^2=17 x^6+56 x^5+27 x^4+8 x^3+11 x^2+6 x+21$
• $y^2=24 x^6+68 x^5+48 x^4+53 x^3+7 x^2+4 x$
• $y^2=8 x^6+72 x^5+21 x^4+13 x^3+51 x^2+51 x+38$
• $y^2=54 x^6+55 x^5+54 x^4+36 x^3+3 x^2+26 x+66$
• $y^2=17 x^6+38 x^5+8 x^4+61 x^3+70 x^2+36 x+13$
• $y^2=4 x^6+63 x^5+6 x^4+34 x^3+39 x^2+64 x+27$
• $y^2=28 x^6+66 x^5+44 x^4+70 x^3+67 x^2+18 x+7$
• $y^2=48 x^6+8 x^5+41 x^4+59 x^3+42 x^2+16 x+1$
• $y^2=55 x^6+37 x^5+51 x^4+48 x^3+8 x^2+41 x+37$
• $y^2=41 x^6+51 x^5+62 x^4+33 x^3+56 x^2+8 x+64$
• $y^2=65 x^6+19 x^5+65 x^4+59 x^3+40 x^2+70 x+72$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73^{3}}$.

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.h $\times$ 1.73.k and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.73.h : \(\Q(\sqrt{-3}) \).
• 1.73.k : \(\Q(\sqrt{-3}) \).

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

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

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.ar_ii	$2$	(not in LMFDB)
2.73.ad_cy	$2$	(not in LMFDB)
2.73.d_cy	$2$	(not in LMFDB)
2.73.abi_qt	$3$	(not in LMFDB)
2.73.ak_bb	$3$	(not in LMFDB)
2.73.ah_ay	$3$	(not in LMFDB)
2.73.o_hn	$3$	(not in LMFDB)
2.73.u_jm	$3$	(not in LMFDB)