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

• Label: 2.53.a_dz
• 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:	$1 + 103 x^{2} + 2809 x^{4}$
Frobenius angles:	$\pm0.462044697231$, $\pm0.537955302769$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{3}, \sqrt{-209})\)
Galois group:	$C_2^2$
Jacobians:	$38$
Isomorphism classes:	40

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2913$	$8485569$	$22164585876$	$62180968311081$	$174887470674500793$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$3016$	$148878$	$7880500$	$418195494$	$22164810622$	$1174711139838$	$62259672153124$	$3299763591802134$	$174887470983488536$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 38 curves (of which all are hyperelliptic):

• $y^2=x^6+x^3+35$
• $y^2=32 x^6+39 x^5+18 x^4+3 x^3+52 x+49$
• $y^2=11 x^6+25 x^5+36 x^4+6 x^3+51 x+45$
• $y^2=14 x^6+11 x^5+3 x^4+15 x^3+19 x^2+16 x+43$
• $y^2=28 x^6+22 x^5+6 x^4+30 x^3+38 x^2+32 x+33$
• $y^2=9 x^6+16 x^5+19 x^4+23 x^3+32 x^2+37 x+10$
• $y^2=18 x^6+32 x^5+38 x^4+46 x^3+11 x^2+21 x+20$
• $y^2=34 x^6+32 x^5+18 x^4+40 x^3+45 x^2+13 x+36$
• $y^2=15 x^6+11 x^5+36 x^4+27 x^3+37 x^2+26 x+19$
• $y^2=41 x^6+33 x^5+3 x^4+36 x^3+36 x^2+52 x+42$
• $y^2=29 x^6+13 x^5+6 x^4+19 x^3+19 x^2+51 x+31$
• $y^2=47 x^6+13 x^5+45 x^4+33 x^3+29 x^2+42 x+34$
• $y^2=41 x^6+26 x^5+37 x^4+13 x^3+5 x^2+31 x+15$
• $y^2=49 x^6+47 x^5+38 x^4+43 x^3+33 x^2+10 x+3$
• $y^2=45 x^6+41 x^5+23 x^4+33 x^3+13 x^2+20 x+6$
• $y^2=29 x^6+14 x^5+10 x^4+15 x^3+6 x^2+48 x+50$
• $y^2=5 x^6+28 x^5+20 x^4+30 x^3+12 x^2+43 x+47$
• $y^2=19 x^6+11 x^5+48 x^4+15 x^3+4 x^2+22 x+15$
• $y^2=38 x^6+22 x^5+43 x^4+30 x^3+8 x^2+44 x+30$
• $y^2=43 x^6+13 x^5+5 x^4+x^3+6 x^2+35 x+13$
• and

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

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 \(\Q(\sqrt{3}, \sqrt{-209})\).

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

The base change of $A$ to $\F_{53^{2}}$ is 1.2809.dz^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-627}) \)$)$

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_adz	$4$	(not in LMFDB)
2.53.ad_ce	$12$	(not in LMFDB)
2.53.d_ce	$12$	(not in LMFDB)