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

• Label: 2.53.aj_eu
• 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:	$( 1 - 6 x + 53 x^{2} )( 1 - 3 x + 53 x^{2} )$
	$1 - 9 x + 124 x^{2} - 477 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.364801829573$, $\pm0.433942022438$
Angle rank:	$2$ (numerical)
Jacobians:	$40$
Isomorphism classes:	200
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2448$	$8372160$	$22341858048$	$62235456019200$	$174858186615626448$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$45$	$2977$	$150066$	$7887409$	$418125465$	$22164209494$	$1174713655797$	$62259707086081$	$3299763554276778$	$174887469586177657$

Jacobians and polarizations

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

• $y^2=21 x^6+43 x^5+18 x^4+27 x^3+39 x^2+26 x+10$
• $y^2=10 x^6+8 x^5+12 x^4+44 x^3+21 x^2+39 x+47$
• $y^2=18 x^6+36 x^5+6 x^4+37 x^3+47 x^2+42 x+11$
• $y^2=45 x^6+38 x^5+33 x^4+35 x^3+18 x^2+28 x+13$
• $y^2=10 x^6+39 x^5+26 x^4+22 x^3+48 x^2+37 x+42$
• $y^2=39 x^6+5 x^5+46 x^4+35 x^3+22 x^2+38 x+13$
• $y^2=21 x^6+26 x^5+15 x^4+47 x^3+44 x^2+50 x+8$
• $y^2=19 x^6+15 x^5+31 x^4+27 x^3+22 x^2+51 x+31$
• $y^2=26 x^6+17 x^5+2 x^4+21 x^3+52 x^2+44 x+7$
• $y^2=39 x^6+14 x^5+37 x^4+27 x^3+20 x^2+18 x+2$
• $y^2=7 x^6+15 x^5+46 x^4+20 x^3+45 x^2+33 x+19$
• $y^2=47 x^6+4 x^5+42 x^4+22 x^3+3 x^2+22 x+27$
• $y^2=16 x^6+2 x^5+35 x^4+22 x^3+7 x^2+30 x+15$
• $y^2=14 x^6+16 x^5+41 x^4+12 x^3+15 x^2+15 x+51$
• $y^2=19 x^6+13 x^5+44 x^4+4 x^3+33 x^2+51 x+15$
• $y^2=47 x^6+8 x^5+51 x^4+35 x^3+x^2+52 x+30$
• $y^2=42 x^6+51 x^5+51 x^4+47 x^3+9 x^2+22 x+22$
• $y^2=20 x^6+x^5+35 x^4+46 x^3+23 x^2+27 x+20$
• $y^2=42 x^6+17 x^5+2 x^4+49 x^3+44 x^2+38 x+16$
• $y^2=27 x^6+12 x^5+16 x^4+32 x^3+47 x^2+34 x+8$
• and

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.ag $\times$ 1.53.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.53.ag : \(\Q(\sqrt{-11}) \).
• 1.53.ad : \(\Q(\sqrt{-203}) \).

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.ad_dk	$2$	(not in LMFDB)
2.53.d_dk	$2$	(not in LMFDB)
2.53.j_eu	$2$	(not in LMFDB)