Abelian variety isogeny class 2.53.a_acz over $\F_{53}$

• Label: 2.53.a_acz
• Base field: $\F_{53}$
• 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:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$1 - 77 x^{2} + 2809 x^{4}$
Frobenius angles:	$\pm0.120592514882$, $\pm0.879407485118$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-29}, \sqrt{183})\)
Galois group:	$C_2^2$
Jacobians:	$30$
Isomorphism classes:	72

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2733$	$7469289$	$22164553476$	$62254798409241$	$174887471032899693$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$2656$	$148878$	$7889860$	$418195494$	$22164745822$	$1174711139838$	$62259721779844$	$3299763591802134$	$174887471700286336$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

• $y^2=9 x^6+36 x^5+29 x^4+30 x^3+26 x^2+29 x+33$
• $y^2=9 x^6+39 x^5+47 x^4+8 x^3+x^2+2 x+4$
• $y^2=18 x^6+25 x^5+41 x^4+16 x^3+2 x^2+4 x+8$
• $y^2=6 x^6+39 x^5+11 x^4+17 x^3+43 x^2+21 x+16$
• $y^2=12 x^6+25 x^5+22 x^4+34 x^3+33 x^2+42 x+32$
• $y^2=23 x^6+x^5+49 x^4+35 x^3+26 x^2+10 x+3$
• $y^2=46 x^6+2 x^5+45 x^4+17 x^3+52 x^2+20 x+6$
• $y^2=9 x^6+4 x^5+49 x^4+22 x^3+9 x^2+31 x+45$
• $y^2=18 x^6+8 x^5+45 x^4+44 x^3+18 x^2+9 x+37$
• $y^2=51 x^6+31 x^5+19 x^4+11 x^3+8 x^2+12 x+22$
• $y^2=49 x^6+9 x^5+38 x^4+22 x^3+16 x^2+24 x+44$
• $y^2=6 x^6+14 x^4+28 x^3+24 x^2+31$
• $y^2=30 x^6+20 x^5+14 x^4+16 x^3+9 x^2+18 x+6$
• $y^2=17 x^6+18 x^5+30 x^4+16 x^3+29 x^2+20 x+32$
• $y^2=24 x^6+38 x^5+51 x^4+25 x^3+24 x^2+50 x+9$
• $y^2=48 x^6+23 x^5+49 x^4+50 x^3+48 x^2+47 x+18$
• $y^2=2 x^6+17 x^5+49 x^4+20 x^3+20 x^2+52 x+22$
• $y^2=4 x^6+34 x^5+45 x^4+40 x^3+40 x^2+51 x+44$
• $y^2=26 x^6+9 x^5+32 x^4+25 x^3+51 x^2+37 x+8$
• $y^2=52 x^6+18 x^5+11 x^4+50 x^3+49 x^2+21 x+16$
• $y^2=23 x^6+22 x^5+32 x^4+27 x^3+50 x^2+44 x+47$
• $y^2=46 x^6+44 x^5+11 x^4+x^3+47 x^2+35 x+41$
• $y^2=12 x^6+16 x^5+6 x^4+33 x^3+8 x^2+52 x+52$
• $y^2=19 x^6+32 x^5+12 x^4+19 x^3+21 x^2+24 x+21$
• $y^2=38 x^6+11 x^5+24 x^4+38 x^3+42 x^2+48 x+42$
• $y^2=49 x^6+2 x^5+46 x^4+40 x^3+23 x^2+x+45$
• $y^2=45 x^6+4 x^5+39 x^4+27 x^3+46 x^2+2 x+37$
• $y^2=12 x^6+44 x^5+44 x^4+3 x^3+35 x^2+17 x+43$
• $y^2=29 x^6+7 x^5+13 x^4+8 x^3+39 x+38$
• $y^2=5 x^6+14 x^5+26 x^4+16 x^3+25 x+23$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53^{2}}$.

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-29}, \sqrt{183})\).

Endomorphism algebra over $\overline{\F}_{53}$

The base change of $A$ to $\F_{53^{2}}$ is 1.2809.acz^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5307}) \)$)$

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
2.53.a_cz	$4$	(not in LMFDB)