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

• Label: 2.73.a_aek
• 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:	$1 - 114 x^{2} + 5329 x^{4}$
Frobenius angles:	$\pm0.107400302387$, $\pm0.892599697613$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{65})\)
Galois group:	$C_2^2$
Jacobians:	$64$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5216$	$27206656$	$151334567264$	$806327363977216$	$4297625833738154336$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$5102$	$389018$	$28393566$	$2073071594$	$151334908238$	$11047398519098$	$806460194554558$	$58871586708267914$	$4297625837772751022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 64 curves (of which all are hyperelliptic):

• $y^2=5 x^6+27 x^5+64 x^4+60 x^3+12 x^2+63 x+39$
• $y^2=35 x^6+50 x^5+54 x^4+9 x^3+16 x^2+9 x+52$
• $y^2=29 x^6+31 x^5+51 x^4+45 x^3+7 x^2+45 x+41$
• $y^2=54 x^6+20 x^5+13 x^4+40 x^3+15 x^2+47 x+54$
• $y^2=51 x^6+27 x^5+65 x^4+54 x^3+2 x^2+16 x+51$
• $y^2=23 x^6+59 x^5+7 x^4+68 x^3+44 x^2+9 x+60$
• $y^2=58 x^6+5 x^5+23 x^4+50 x^3+53 x^2+30 x+37$
• $y^2=64 x^6+62 x^5+52 x^4+5 x^3+62 x^2+70 x+27$
• $y^2=28 x^6+18 x^5+41 x^4+25 x^3+18 x^2+58 x+62$
• $y^2=31 x^6+18 x^5+40 x^4+11 x^3+36 x^2+28 x+24$
• $y^2=9 x^6+17 x^5+54 x^4+55 x^3+34 x^2+67 x+47$
• $y^2=x^6+48 x^5+15 x^4+57 x^3+52 x^2+66 x+52$
• $y^2=5 x^6+21 x^5+2 x^4+66 x^3+41 x^2+38 x+41$
• $y^2=21 x^6+67 x^5+63 x^4+33 x^3+17 x^2+24 x+5$
• $y^2=44 x^6+55 x^5+37 x^4+37 x^3+72 x^2+25 x+66$
• $y^2=26 x^6+69 x^5+13 x^4+29 x^3+37 x^2+19 x+50$
• $y^2=47 x^6+7 x^5+61 x^4+46 x^3+47 x+11$
• $y^2=16 x^6+35 x^5+13 x^4+11 x^3+16 x+55$
• $y^2=68 x^6+20 x^5+55 x^4+8 x^3+12 x^2+22 x+71$
• $y^2=48 x^6+27 x^5+56 x^4+40 x^3+60 x^2+37 x+63$
• and

• $y^2=5 x^6+38 x^5+43 x^4+43 x^3+71 x^2+55 x+69$
• $y^2=28 x^6+58 x^5+5 x^4+18 x^3+22 x^2+65 x+60$
• $y^2=67 x^6+71 x^5+25 x^4+17 x^3+37 x^2+33 x+8$
• $y^2=13 x^6+10 x^5+17 x^4+67 x^3+65 x^2+30 x+16$
• $y^2=4 x^6+46 x^5+69 x^4+40 x^3+50 x^2+21 x+6$
• $y^2=20 x^6+11 x^5+53 x^4+54 x^3+31 x^2+32 x+30$
• $y^2=46 x^6+21 x^5+20 x^3+17 x+22$
• $y^2=9 x^6+14 x^5+10 x^4+21 x^2+55 x+28$
• $y^2=28 x^6+22 x^5+41 x^4+69 x^3+41 x^2+6 x+37$
• $y^2=29 x^6+15 x^5+54 x^4+58 x^3+22 x^2+23 x+18$
• $y^2=72 x^6+2 x^5+51 x^4+71 x^3+37 x^2+42 x+17$
• $y^2=67 x^6+51 x^5+7 x^4+15 x^3+55 x^2+5 x+42$
• $y^2=60 x^6+70 x^5+63 x^4+13 x^3+29 x^2+4 x+50$
• $y^2=63 x^6+49 x^5+57 x^4+25 x^3+4 x^2+20 x+33$
• $y^2=23 x^6+26 x^5+66 x^4+52 x^3+20 x^2+27 x+19$
• $y^2=9 x^6+41 x^5+43 x^4+70 x^3+54 x^2+72 x+62$
• $y^2=45 x^6+59 x^5+69 x^4+58 x^3+51 x^2+68 x+18$
• $y^2=65 x^6+72 x^5+16 x^4+61 x^3+23 x^2+36 x+51$
• $y^2=50 x^6+42 x^5+31 x^4+27 x^3+3 x^2+38 x+62$
• $y^2=45 x^6+12 x^5+39 x^4+16 x^3+13 x^2+39 x+22$
• $y^2=6 x^6+60 x^5+49 x^4+7 x^3+65 x^2+49 x+37$
• $y^2=57 x^6+47 x^5+71 x^4+10 x^3+25 x^2+59 x+38$
• $y^2=69 x^6+38 x^5+64 x^4+70 x^3+6 x^2+16 x+29$
• $y^2=53 x^6+44 x^5+28 x^4+58 x^3+30 x^2+7 x+72$
• $y^2=22 x^6+35 x^5+15 x^4+37 x^3+66 x^2+24 x+32$
• $y^2=37 x^6+29 x^5+2 x^4+39 x^3+38 x^2+47 x+14$
• $y^2=34 x^6+44 x^5+62 x^4+17 x^3+35 x^2+34 x+25$
• $y^2=58 x^6+5 x^5+19 x^4+45 x^3+57 x^2+71 x+34$
• $y^2=41 x^6+46 x^5+52 x^4+20 x^3+57 x^2+8 x+57$
• $y^2=59 x^6+11 x^5+41 x^4+27 x^3+66 x^2+40 x+66$
• $y^2=37 x^6+37 x^5+67 x^4+39 x^3+12 x^2+2 x+55$
• $y^2=32 x^6+68 x^5+21 x^4+49 x^3+69 x^2+22 x+15$
• $y^2=61 x^6+63 x^5+58 x^4+60 x^3+52 x^2+9 x+49$
• $y^2=13 x^6+23 x^5+71 x^4+8 x^3+41 x^2+45 x+26$
• $y^2=14 x^6+16 x^5+63 x^4+47 x^3+58 x^2+57 x+49$
• $y^2=70 x^6+7 x^5+23 x^4+16 x^3+71 x^2+66 x+26$
• $y^2=34 x^6+8 x^5+34 x^4+20 x^3+65 x^2+6 x+67$
• $y^2=3 x^6+3 x^5+x^4+32 x^3+63 x^2+62 x+18$
• $y^2=15 x^6+15 x^5+5 x^4+14 x^3+23 x^2+18 x+17$
• $y^2=34 x^6+10 x^5+69 x^4+36 x^3+33 x^2+51 x+55$
• $y^2=8 x^6+68 x^5+49 x^4+44 x^3+42 x^2+38 x+23$
• $y^2=38 x^6+66 x^5+30 x^4+60 x^3+7 x^2+40 x+69$
• $y^2=24 x^6+3 x^5+31 x^4+71 x^3+61 x^2+61 x+60$
• $y^2=47 x^6+15 x^5+9 x^4+63 x^3+13 x^2+13 x+8$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73^{2}}$.

Endomorphism algebra over $\F_{73}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{65})\).

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

The base change of $A$ to $\F_{73^{2}}$ is 1.5329.aek^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-130}) \)$)$

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.a_ek	$4$	(not in LMFDB)
2.73.ai_bg	$8$	(not in LMFDB)
2.73.i_bg	$8$	(not in LMFDB)