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

• Label: 2.73.m_ct
• 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:	$1 + 12 x + 71 x^{2} + 876 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.414486393774$, $\pm0.918846939559$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-37})\)
Galois group:	$C_2^2$
Jacobians:	$48$
Isomorphism classes:	52

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6289$	$28382257$	$152036046724$	$806157622133289$	$4297496994239436049$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$86$	$5328$	$390818$	$28387588$	$2073009446$	$151334162358$	$11047403439494$	$806460148605316$	$58871586065576114$	$4297625829419788368$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

• $y^2=65 x^6+8 x^5+12 x^4+66 x^3+47 x^2+7 x+1$
• $y^2=35 x^6+42 x^5+13 x^4+58 x^3+47 x^2+46 x+27$
• $y^2=66 x^6+21 x^5+27 x^4+50 x^2+40 x+56$
• $y^2=67 x^6+57 x^5+61 x^4+53 x^3+55 x^2+49 x+6$
• $y^2=34 x^6+13 x^5+14 x^4+50 x^3+34 x^2+x+43$
• $y^2=51 x^6+70 x^5+9 x^4+22 x^3+41 x^2+24 x+41$
• $y^2=48 x^6+37 x^5+69 x^4+46 x^3+13 x^2+58 x+5$
• $y^2=12 x^6+28 x^5+15 x^4+58 x^3+39 x^2+24 x+17$
• $y^2=48 x^6+5 x^5+5 x^4+24 x^3+44 x^2+31 x+41$
• $y^2=26 x^6+46 x^5+42 x^4+37 x^3+34 x^2+24$
• $y^2=67 x^6+33 x^5+27 x^4+62 x^3+38 x^2+55 x+21$
• $y^2=29 x^6+47 x^5+44 x^4+39 x^3+51 x^2+23 x+25$
• $y^2=61 x^6+2 x^5+16 x^4+50 x^3+36 x^2+9 x+18$
• $y^2=28 x^6+5 x^5+66 x^4+67 x^3+28 x^2+24 x+32$
• $y^2=55 x^6+41 x^5+53 x^4+4 x^3+53 x^2+64 x+19$
• $y^2=28 x^6+x^5+16 x^4+63 x^3+47 x^2+23 x+66$
• $y^2=61 x^6+40 x^5+28 x^4+59 x^3+69 x^2+62 x+38$
• $y^2=34 x^6+39 x^5+7 x^4+51 x^3+50 x^2+26 x+35$
• $y^2=61 x^6+55 x^4+52 x^3+33 x^2+31 x+19$
• $y^2=57 x^6+28 x^5+58 x^4+72 x^3+43 x^2+60 x+32$
• and

• $y^2=68 x^6+55 x^5+6 x^4+48 x^3+60 x^2+17 x+8$
• $y^2=18 x^6+33 x^5+23 x^4+36 x^3+31 x^2+63 x+22$
• $y^2=30 x^6+26 x^5+13 x^4+36 x^3+49 x^2+32 x+25$
• $y^2=36 x^6+65 x^5+8 x^4+23 x^3+x^2+8 x+30$
• $y^2=43 x^6+39 x^5+64 x^4+34 x^3+17 x^2+23 x+33$
• $y^2=34 x^6+50 x^5+57 x^4+12 x^3+59 x^2+13 x+50$
• $y^2=68 x^6+66 x^5+38 x^4+29 x^3+33 x^2+43 x+24$
• $y^2=46 x^6+30 x^5+63 x^4+66 x^3+54 x^2+71 x+14$
• $y^2=56 x^6+11 x^5+13 x^4+48 x^3+4 x^2+62 x+53$
• $y^2=61 x^6+3 x^5+7 x^4+34 x^3+72 x^2+35 x+4$
• $y^2=66 x^6+30 x^5+33 x^4+16 x^3+61 x^2+28 x+42$
• $y^2=38 x^6+65 x^5+61 x^4+10 x^3+10 x^2+28 x+49$
• $y^2=4 x^6+46 x^5+44 x^3+44 x^2+11 x+68$
• $y^2=56 x^6+33 x^5+38 x^4+49 x^3+5 x^2+45 x+69$
• $y^2=66 x^6+11 x^5+31 x^4+48 x^3+63 x^2+2 x+70$
• $y^2=40 x^6+48 x^5+28 x^3+69 x^2+4 x+51$
• $y^2=25 x^6+39 x^5+7 x^4+72 x^3+56 x^2+4 x+12$
• $y^2=2 x^6+53 x^5+7 x^4+20 x^3+65 x^2+55 x+8$
• $y^2=5 x^6+18 x^5+43 x^4+67 x^3+21 x^2+57 x+71$
• $y^2=12 x^6+4 x^5+27 x^4+68 x^3+44 x^2+43 x+36$
• $y^2=22 x^6+56 x^5+69 x^4+12 x^3+34 x^2+68 x+13$
• $y^2=29 x^6+20 x^5+49 x^4+43 x^3+53 x^2+6$
• $y^2=50 x^6+30 x^5+46 x^4+12 x^3+30 x^2+43 x+52$
• $y^2=8 x^6+19 x^5+19 x^4+63 x^3+45 x^2+66 x+69$
• $y^2=50 x^6+38 x^5+55 x^4+33 x^3+43 x^2+53$
• $y^2=45 x^6+68 x^5+23 x^4+67 x^3+33 x^2+14 x+72$
• $y^2=60 x^6+60 x^5+2 x^4+33 x^3+57 x^2+4 x+6$
• $y^2=4 x^6+7 x^5+16 x^4+53 x^3+17 x^2+58 x+13$

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 \(\Q(\sqrt{-3}, \sqrt{-37})\).

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

The base change of $A$ to $\F_{73^{3}}$ is 1.389017.biq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-37}) \)$)$

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.am_ct	$2$	(not in LMFDB)
2.73.ay_le	$3$	(not in LMFDB)
2.73.a_c	$6$	(not in LMFDB)