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

• Label: 2.53.av_ia
• Base field: $\F_{53}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$1 - 21 x + 208 x^{2} - 1113 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.129472067798$, $\pm0.324486313334$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-374 +42 \sqrt{33}})\)
Galois group:	$D_{4}$
Jacobians:	$36$
Isomorphism classes:	36
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1884$	$7822368$	$22239489600$	$62288483831424$	$174888691634249004$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$33$	$2785$	$149382$	$7894129$	$418198413$	$22164295030$	$1174711787961$	$62259710775169$	$3299763802863726$	$174887471469170425$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

• $y^2=8 x^6+x^5+45 x^4+30 x^3+48 x^2+21 x+21$
• $y^2=7 x^6+5 x^5+40 x^4+13 x^3+8 x^2+16 x+21$
• $y^2=48 x^6+9 x^5+10 x^4+43 x^3+16 x^2+17 x$
• $y^2=x^6+47 x^5+20 x^4+23 x^3+50 x^2+38 x+48$
• $y^2=41 x^6+14 x^5+18 x^4+17 x^3+43 x^2+2 x+16$
• $y^2=18 x^6+32 x^5+31 x^4+39 x^3+19 x^2+22 x+33$
• $y^2=27 x^6+7 x^5+48 x^4+48 x^3+30 x^2+49 x+37$
• $y^2=39 x^6+9 x^5+49 x^4+49 x^3+42 x^2+12 x+32$
• $y^2=51 x^6+14 x^5+52 x^4+3 x^3+39 x^2+34 x+36$
• $y^2=20 x^6+41 x^5+28 x^4+15 x^3+50 x^2+32$
• $y^2=26 x^6+52 x^5+42 x^4+17 x^3+33 x^2+22 x$
• $y^2=34 x^6+24 x^5+51 x^4+4 x^3+20 x^2+46 x+14$
• $y^2=21 x^6+30 x^5+17 x^4+33 x^3+52 x^2+3 x+11$
• $y^2=9 x^6+50 x^5+49 x^4+12 x^3+6 x^2+6 x+27$
• $y^2=50 x^6+27 x^5+34 x^4+5 x^3+18 x^2+52 x+48$
• $y^2=21 x^6+43 x^5+25 x^4+49 x^3+17 x^2+12 x+48$
• $y^2=20 x^6+19 x^5+17 x^4+29 x^3+51 x^2+13 x+30$
• $y^2=36 x^6+28 x^5+45 x^4+37 x^3+5 x^2+46 x+23$
• $y^2=45 x^6+7 x^5+20 x^4+41 x^3+37 x^2+24 x+26$
• $y^2=28 x^6+3 x^5+14 x^4+47 x^3+28 x+22$
• and

• $y^2=27 x^6+39 x^5+13 x^4+4 x^3+8 x^2+43 x$
• $y^2=33 x^6+30 x^5+17 x^4+48 x^3+4 x^2+29 x+28$
• $y^2=24 x^6+4 x^5+30 x^4+20 x^3+51 x^2+40 x+45$
• $y^2=30 x^6+46 x^5+45 x^4+6 x^3+6 x^2+30$
• $y^2=51 x^6+27 x^5+28 x^4+9 x^3+16 x^2+12 x+1$
• $y^2=38 x^6+52 x^5+25 x^4+23 x^3+28 x^2+46 x+33$
• $y^2=17 x^5+46 x^3+30 x^2+44 x+19$
• $y^2=22 x^6+41 x^5+6 x^4+37 x^3+15 x^2+32 x+32$
• $y^2=32 x^6+8 x^5+50 x^4+10 x^3+50 x^2+x+38$
• $y^2=42 x^6+27 x^5+14 x^4+38 x^3+14 x^2+x+19$
• $y^2=22 x^6+31 x^5+14 x^4+38 x^3+44 x^2+18 x+45$
• $y^2=51 x^6+19 x^5+33 x^4+38 x^3+x^2+27 x+20$
• $y^2=5 x^6+35 x^5+17 x^4+30 x^3+19 x^2+45 x+26$
• $y^2=8 x^6+6 x^5+11 x^4+3 x^3+30 x^2+48 x+8$
• $y^2=25 x^6+30 x^5+52 x^4+51 x^3+14 x^2+31 x+2$
• $y^2=47 x^6+14 x^5+8 x^4+42 x^3+4 x^2+3 x+15$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-374 +42 \sqrt{33}})\).

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.v_ia	$2$	(not in LMFDB)