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

• Label: 2.53.b_aca
• 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:	$1 + x - 52 x^{2} + 53 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.188545502792$, $\pm0.855212169458$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-211})\)
Galois group:	$C_2^2$
Jacobians:	$39$
Isomorphism classes:	69

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2812$	$7603648$	$22211729296$	$62302375592704$	$174893233289336332$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$55$	$2705$	$149194$	$7895889$	$418209275$	$22164906710$	$1174710136655$	$62259703866049$	$3299763458555362$	$174887469719038025$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 39 curves (of which all are hyperelliptic):

• $y^2=8 x^6+52 x^5+42 x^4+34 x^3+12 x^2+6 x+15$
• $y^2=18 x^6+47 x^5+3 x^4+50 x^3+13 x^2+18 x+30$
• $y^2=x^6+27 x^5+18 x^4+20 x^3+24 x^2+26 x+23$
• $y^2=42 x^6+9 x^5+14 x^4+42 x^3+37 x^2+7 x+27$
• $y^2=30 x^6+8 x^5+7 x^4+43 x^3+50 x^2+40 x+16$
• $y^2=43 x^6+33 x^5+42 x^4+5 x^3+43 x^2+5 x+29$
• $y^2=x^6+46 x^5+48 x^4+16 x^3+12 x^2+44 x+3$
• $y^2=4 x^6+42 x^5+9 x^4+41 x^3+3 x^2+12 x+21$
• $y^2=36 x^6+19 x^5+17 x^4+44 x^3+47 x^2+23 x$
• $y^2=42 x^6+43 x^5+14 x^4+4 x^3+15 x^2+x+40$
• $y^2=29 x^6+7 x^5+41 x^4+35 x^3+9 x^2+15 x+2$
• $y^2=50 x^6+17 x^5+42 x^4+46 x^3+40 x^2+37 x+21$
• $y^2=10 x^6+49 x^5+4 x^4+9 x^3+23 x^2+35 x$
• $y^2=32 x^6+3 x^5+11 x^4+51 x^3+43 x^2+23 x+2$
• $y^2=19 x^6+13 x^5+37 x^4+7 x^3+40 x^2+10 x+25$
• $y^2=12 x^6+26 x^5+38 x^4+15 x^3+3 x^2+36 x$
• $y^2=36 x^6+24 x^5+47 x^4+24 x^3+37 x^2+20 x+9$
• $y^2=31 x^6+39 x^5+7 x^4+13 x^3+39 x+14$
• $y^2=23 x^6+12 x^5+18 x^4+19 x^3+30 x^2+17 x+37$
• $y^2=4 x^6+21 x^5+44 x^4+44 x^3+39 x^2+31 x+38$
• and

• $y^2=37 x^6+4 x^5+43 x^4+8 x^3+44 x^2+31 x+24$
• $y^2=33 x^6+39 x^5+19 x^4+43 x^3+27 x^2+27 x+17$
• $y^2=14 x^6+24 x^5+27 x^4+24 x^3+48 x^2+39 x+47$
• $y^2=23 x^6+46 x^5+47 x^4+44 x^3+24 x^2+7 x+43$
• $y^2=49 x^6+29 x^5+7 x^4+10 x^3+33 x^2+31 x+34$
• $y^2=40 x^6+50 x^5+48 x^4+23 x^3+6 x^2+22 x$
• $y^2=42 x^6+45 x^5+26 x^4+40 x^3+36 x^2+38 x+34$
• $y^2=40 x^6+41 x^5+22 x^4+18 x^3+30 x^2+39 x+5$
• $y^2=21 x^6+4 x^5+31 x^4+2 x^3+16 x^2+13 x$
• $y^2=3 x^6+47 x^5+39 x^4+26 x^3+39 x^2+41 x+24$
• $y^2=33 x^6+21 x^5+31 x^4+45 x^3+27 x^2+30 x+51$
• $y^2=19 x^6+22 x^5+21 x^4+8 x^3+52 x^2+33 x+38$
• $y^2=12 x^6+33 x^5+4 x^4+22 x^3+18 x^2+9 x+20$
• $y^2=30 x^5+x^4+7 x^3+36 x^2+38 x+43$
• $y^2=41 x^6+34 x^5+7 x^4+47 x^3+7 x^2+51 x+52$
• $y^2=40 x^6+48 x^5+50 x^4+3 x^3+12 x^2+41$
• $y^2=51 x^6+49 x^5+15 x^4+47 x^3+27 x^2+4 x+26$
• $y^2=37 x^6+5 x^5+29 x^4+30 x^3+15 x^2+4 x+39$
• $y^2=21 x^6+42 x^5+42 x^4+9 x^3+7 x^2+24 x+5$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53^{3}}$.

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-211})\).

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

The base change of $A$ to $\F_{53^{3}}$ is 1.148877.gc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-211}) \)$)$

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.ab_aca	$2$	(not in LMFDB)
2.53.ac_ed	$3$	(not in LMFDB)
2.53.a_eb	$6$	(not in LMFDB)