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

• Label: 2.53.as_hf
• Base field: $\F_{53}$
• 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:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x + 53 x^{2} )^{2}$
	$1 - 18 x + 187 x^{2} - 954 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.287893547303$, $\pm0.287893547303$
Angle rank:	$1$ (numerical)
Jacobians:	$40$
Cyclic group of points:	no
Non-cyclic primes:	$3, 5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2025$	$8037225$	$22374176400$	$62338525475625$	$174893937400400625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$2860$	$150282$	$7900468$	$418210956$	$22163971030$	$1174706809452$	$62259672113188$	$3299763656629026$	$174887471918758300$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

• $y^2=19 x^6+29 x^5+25 x^4+41 x^3+42 x^2+52 x+22$
• $y^2=39 x^6+10 x^5+25 x^4+7 x^3+44 x^2+42 x+50$
• $y^2=19 x^6+28 x^5+48 x^4+6 x^3+19 x^2+10 x+31$
• $y^2=33 x^6+52 x^5+14 x^4+18 x^3+18 x^2+7 x+21$
• $y^2=21 x^6+27 x^5+36 x^4+4 x^3+10 x^2+33 x+14$
• $y^2=29 x^6+18 x^5+51 x^4+40 x^3+30 x^2+22 x+16$
• $y^2=7 x^6+6 x^5+36 x^4+46 x^3+52 x^2+2 x+24$
• $y^2=22 x^6+19 x^5+13 x^4+9 x^3+28 x^2+48 x+18$
• $y^2=26 x^6+50 x^5+34 x^4+47 x^3+19 x^2+50 x+27$
• $y^2=51 x^6+23 x^5+15 x^4+46 x^3+7 x^2+50 x+31$
• $y^2=45 x^6+17 x^5+21 x^4+37 x^3+50 x^2+9 x+30$
• $y^2=22 x^6+12 x^5+49 x^4+42 x^3+18 x^2+4 x+3$
• $y^2=7 x^6+23 x^5+23 x^4+26 x^3+34 x^2+18 x+9$
• $y^2=7 x^6+10 x^5+35 x^4+44 x^3+5 x^2+24 x+13$
• $y^2=8 x^6+27 x^5+23 x^4+43 x^3+39 x^2+8 x+35$
• $y^2=5 x^6+31 x^5+32 x^4+27 x^3+21 x^2+31 x+48$
• $y^2=30 x^6+38 x^5+37 x^4+50 x^3+44 x^2+29 x+27$
• $y^2=51 x^6+46 x^4+18 x^3+44 x^2+18$
• $y^2=26 x^6+11 x^5+35 x^4+44 x^3+12 x^2+52 x+8$
• $y^2=8 x^6+43 x^5+38 x^4+39 x^3+38 x^2+43 x+8$
• and

• $y^2=28 x^6+13 x^5+8 x^4+24 x^3+31 x^2+9 x+7$
• $y^2=25 x^6+50 x^5+33 x^4+30 x^2+33 x+15$
• $y^2=43 x^6+8 x^5+13 x^4+27 x^3+4 x^2+34 x+44$
• $y^2=5 x^6+44 x^5+16 x^4+18 x^3+10 x^2+47 x+33$
• $y^2=50 x^6+51 x^5+11 x^4+37 x^3+52 x^2+14 x+27$
• $y^2=26 x^6+9 x^5+45 x^4+39 x^3+23 x^2+4 x+8$
• $y^2=8 x^6+50 x^5+46 x^4+10 x^3+30 x^2+15 x+18$
• $y^2=22 x^6+27 x^5+24 x^4+43 x^3+11 x^2+33 x+3$
• $y^2=18 x^6+49 x^5+52 x^4+38 x^3+4 x^2+42 x+14$
• $y^2=40 x^6+18 x^5+2 x^4+19 x^3+33 x^2+51 x+15$
• $y^2=26 x^6+24 x^5+28 x^4+4 x^3+46 x^2+28 x+41$
• $y^2=36 x^6+37 x^5+34 x^4+8 x^3+30 x^2+9 x+25$
• $y^2=10 x^6+9 x^5+36 x^4+51 x^3+44 x^2+37 x+28$
• $y^2=43 x^6+10 x^5+50 x^4+25 x^3+3 x^2+10 x+10$
• $y^2=20 x^6+29 x^5+16 x^4+34 x^3+52 x^2+38 x+34$
• $y^2=45 x^6+40 x^5+50 x^4+49 x^3+3 x^2+40 x+8$
• $y^2=2 x^6+4 x^5+52 x^4+49 x^3+42 x^2+7 x+12$
• $y^2=18 x^6+18 x^5+13 x^4+37 x^3+13 x^2+18 x+18$
• $y^2=31 x^6+23 x^5+36 x^4+32 x^3+46 x^2+5 x+41$
• $y^2=31 x^6+19 x^5+33 x^4+27 x^3+29 x^2+35 x+35$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.aj^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-131}) \)$)$

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
2.53.a_z	$2$	(not in LMFDB)
2.53.s_hf	$2$	(not in LMFDB)
2.53.j_bc	$3$	(not in LMFDB)