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

• Label: 2.73.a_fp
• 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:	$( 1 - x + 73 x^{2} )( 1 + x + 73 x^{2} )$
	$1 + 145 x^{2} + 5329 x^{4}$
Frobenius angles:	$\pm0.481361674224$, $\pm0.518638325776$
Angle rank:	$1$ (numerical)
Jacobians:	$44$
Isomorphism classes:	64
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5475$	$29975625$	$151334956800$	$805871447015625$	$4297625833159009875$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$74$	$5620$	$389018$	$28377508$	$2073071594$	$151335687310$	$11047398519098$	$806459990537668$	$58871586708267914$	$4297625836614462100$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 44 curves (of which all are hyperelliptic):

• $y^2=40 x^6+38 x^5+13 x^4+72 x^3+13 x^2+38 x+40$
• $y^2=54 x^6+44 x^5+65 x^4+68 x^3+65 x^2+44 x+54$
• $y^2=69 x^6+41 x^5+57 x^4+7 x^3+57 x^2+41 x+69$
• $y^2=53 x^6+59 x^5+66 x^4+35 x^3+66 x^2+59 x+53$
• $y^2=25 x^6+43 x^5+66 x^4+36 x^3+66 x^2+43 x+25$
• $y^2=52 x^6+69 x^5+38 x^4+34 x^3+38 x^2+69 x+52$
• $y^2=6 x^6+20 x^5+57 x^4+71 x^3+57 x^2+20 x+6$
• $y^2=30 x^6+27 x^5+66 x^4+63 x^3+66 x^2+27 x+30$
• $y^2=24 x^6+11 x^5+13 x^4+42 x^3+13 x^2+11 x+24$
• $y^2=47 x^6+55 x^5+65 x^4+64 x^3+65 x^2+55 x+47$
• $y^2=43 x^6+56 x^5+5 x^4+15 x^3+5 x^2+56 x+43$
• $y^2=69 x^6+61 x^5+25 x^4+2 x^3+25 x^2+61 x+69$
• $y^2=43 x^6+24 x^5+28 x^4+52 x^3+28 x^2+24 x+43$
• $y^2=69 x^6+47 x^5+67 x^4+41 x^3+67 x^2+47 x+69$
• $y^2=15 x^6+70 x^5+40 x^4+48 x^3+40 x^2+70 x+15$
• $y^2=2 x^6+58 x^5+54 x^4+21 x^3+54 x^2+58 x+2$
• $y^2=41 x^6+12 x^5+69 x^4+36 x^3+69 x^2+12 x+41$
• $y^2=59 x^6+60 x^5+53 x^4+34 x^3+53 x^2+60 x+59$
• $y^2=64 x^6+54 x^5+38 x^4+34 x^3+38 x^2+54 x+64$
• $y^2=28 x^6+51 x^5+44 x^4+24 x^3+44 x^2+51 x+28$
• and

• $y^2=14 x^6+9 x^5+72 x^4+52 x^3+72 x^2+9 x+14$
• $y^2=70 x^6+45 x^5+68 x^4+41 x^3+68 x^2+45 x+70$
• $y^2=71 x^6+57 x^5+42 x^4+17 x^3+42 x^2+57 x+71$
• $y^2=63 x^6+66 x^5+64 x^4+12 x^3+64 x^2+66 x+63$
• $y^2=56 x^6+29 x^5+30 x^4+8 x^3+30 x^2+29 x+56$
• $y^2=61 x^6+72 x^5+4 x^4+40 x^3+4 x^2+72 x+61$
• $y^2=59 x^6+23 x^5+26 x^4+7 x^3+26 x^2+23 x+59$
• $y^2=3 x^6+42 x^5+57 x^4+35 x^3+57 x^2+42 x+3$
• $y^2=11 x^6+47 x^5+24 x^4+6 x^3+24 x^2+47 x+11$
• $y^2=55 x^6+16 x^5+47 x^4+30 x^3+47 x^2+16 x+55$
• $y^2=44 x^6+57 x^5+x^4+39 x^3+x^2+57 x+44$
• $y^2=x^6+66 x^5+5 x^4+49 x^3+5 x^2+66 x+1$
• $y^2=71 x^6+31 x^5+40 x^3+31 x+71$
• $y^2=63 x^6+9 x^5+54 x^3+9 x+63$
• $y^2=4 x^6+56 x^5+28 x^4+42 x^3+28 x^2+56 x+4$
• $y^2=20 x^6+61 x^5+67 x^4+64 x^3+67 x^2+61 x+20$
• $y^2=13 x^6+39 x^5+57 x^4+25 x^3+21 x^2+55 x+46$
• $y^2=65 x^6+49 x^5+66 x^4+52 x^3+32 x^2+56 x+11$
• $y^2=26 x^6+43 x^5+46 x^4+4 x^3+44 x^2+47 x+64$
• $y^2=57 x^6+69 x^5+11 x^4+20 x^3+x^2+16 x+28$
• $y^2=30 x^6+37 x^5+22 x^4+40 x^3+22 x^2+37 x+30$
• $y^2=4 x^6+39 x^5+37 x^4+54 x^3+37 x^2+39 x+4$
• $y^2=52 x^6+17 x^5+4 x^4+27 x^3+4 x^2+17 x+52$
• $y^2=41 x^6+12 x^5+20 x^4+62 x^3+20 x^2+12 x+41$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.ab $\times$ 1.73.b 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.ab : \(\Q(\sqrt{-291}) \).
• 1.73.b : \(\Q(\sqrt{-291}) \).

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

The base change of $A$ to $\F_{73^{2}}$ is 1.5329.fp^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-291}) \)$)$

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.ac_fr	$2$	(not in LMFDB)
2.73.c_fr	$2$	(not in LMFDB)
2.73.a_afp	$4$	(not in LMFDB)