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

• Label: 2.53.a_a
• Base field: $\F_{53}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$1 + 2809 x^{4}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.750000000000$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(i, \sqrt{106})\)
Galois group:	$C_2^2$
Jacobians:	$81$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2810$	$7896100$	$22164361130$	$62348395210000$	$174887470365513050$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$2810$	$148878$	$7901718$	$418195494$	$22164361130$	$1174711139838$	$62259658849438$	$3299763591802134$	$174887470365513050$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 81 curves (of which all are hyperelliptic):

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

• $y^2=6 x^6+34 x^5+48 x^4+x^3+14 x^2+14 x+43$
• $y^2=12 x^6+15 x^5+43 x^4+2 x^3+28 x^2+28 x+33$
• $y^2=9 x^6+27 x^5+48 x^4+36 x^3+17 x^2+10 x+1$
• $y^2=18 x^6+x^5+43 x^4+19 x^3+34 x^2+20 x+2$
• $y^2=9 x^6+42 x^5+27 x^4+49 x^3+13 x^2+40$
• $y^2=18 x^6+31 x^5+x^4+45 x^3+26 x^2+27$
• $y^2=4 x^6+46 x^5+2 x^4+47 x^3+28 x^2+49 x+51$
• $y^2=8 x^6+39 x^5+4 x^4+41 x^3+3 x^2+45 x+49$
• $y^2=49 x^6+45 x^5+41 x^4+28 x^3+40 x^2+41 x+21$
• $y^2=45 x^6+37 x^5+29 x^4+3 x^3+27 x^2+29 x+42$
• $y^2=26 x^6+18 x^5+46 x^4+25 x^3+34 x^2+19 x+31$
• $y^2=52 x^6+36 x^5+39 x^4+50 x^3+15 x^2+38 x+9$
• $y^2=46 x^6+5 x^5+24 x^4+46 x^3+27 x^2+31 x+48$
• $y^2=39 x^6+10 x^5+48 x^4+39 x^3+x^2+9 x+43$
• $y^2=32 x^5+16 x^4+15 x^3+39 x^2+13 x+26$
• $y^2=11 x^5+32 x^4+30 x^3+25 x^2+26 x+52$
• $y^2=27 x^6+37 x^5+25 x^4+19 x^3+46 x^2+19 x+45$
• $y^2=x^6+21 x^5+50 x^4+38 x^3+39 x^2+38 x+37$
• $y^2=x^5+51 x$
• $y^2=13 x^6+x^5+44 x^4+46 x^2+5 x+36$
• $y^2=26 x^6+2 x^5+35 x^4+39 x^2+10 x+19$
• $y^2=47 x^6+43 x^5+25 x^4+6 x^3+43 x^2+41 x+20$
• $y^2=41 x^6+33 x^5+50 x^4+12 x^3+33 x^2+29 x+40$
• $y^2=31 x^6+4 x^5+33 x^4+46 x^3+10 x^2+39 x+36$
• $y^2=9 x^6+8 x^5+13 x^4+39 x^3+20 x^2+25 x+19$
• $y^2=48 x^6+29 x^5+2 x^4+45 x^3+6 x^2+10 x+12$
• $y^2=43 x^6+5 x^5+4 x^4+37 x^3+12 x^2+20 x+24$
• $y^2=13 x^6+48 x^5+42 x^4+40 x^3+36 x^2+26 x+8$
• $y^2=26 x^6+43 x^5+31 x^4+27 x^3+19 x^2+52 x+16$
• $y^2=22 x^6+45 x^5+30 x^4+42 x^3+11 x^2+x+24$
• $y^2=44 x^6+37 x^5+7 x^4+31 x^3+22 x^2+2 x+48$
• $y^2=15 x^6+30 x^5+46 x^4+11 x^3+17 x^2+32 x+10$
• $y^2=30 x^6+7 x^5+39 x^4+22 x^3+34 x^2+11 x+20$
• $y^2=4 x^6+42 x^5+50 x^4+14 x^3+31 x^2+48 x+29$
• $y^2=8 x^6+31 x^5+47 x^4+28 x^3+9 x^2+43 x+5$
• $y^2=39 x^6+29 x^5+6 x^4+12 x^3+48 x^2+44 x+9$
• $y^2=25 x^6+5 x^5+12 x^4+24 x^3+43 x^2+35 x+18$
• $y^2=13 x^6+35 x^5+14 x^4+34 x^3+9 x^2+42 x+5$
• $y^2=26 x^6+17 x^5+28 x^4+15 x^3+18 x^2+31 x+10$
• $y^2=26 x^6+44 x^5+26 x^4+6 x^3+34 x^2+40 x+39$
• $y^2=52 x^6+35 x^5+52 x^4+12 x^3+15 x^2+27 x+25$
• $y^2=38 x^6+24 x^5+28 x^4+39 x^3+38 x^2+26 x+39$
• $y^2=23 x^6+48 x^5+3 x^4+25 x^3+23 x^2+52 x+25$
• $y^2=42 x^6+6 x^5+25 x^4+28 x^3+48 x^2+11 x+32$
• $y^2=31 x^6+12 x^5+50 x^4+3 x^3+43 x^2+22 x+11$
• $y^2=20 x^6+30 x^5+38 x^4+42 x^3+44 x^2+20 x+22$
• $y^2=40 x^6+7 x^5+23 x^4+31 x^3+35 x^2+40 x+44$
• $y^2=35 x^6+8 x^5+9 x^4+20 x^3+31 x^2+10 x+37$
• $y^2=17 x^6+16 x^5+18 x^4+40 x^3+9 x^2+20 x+21$
• $y^2=50 x^6+6 x^5+21 x^4+11 x^3+21 x^2+31$
• $y^2=47 x^6+12 x^5+42 x^4+22 x^3+42 x^2+9$
• $y^2=19 x^6+15 x^5+21 x^4+39 x^3+26 x^2+20 x+8$
• $y^2=38 x^6+30 x^5+42 x^4+25 x^3+52 x^2+40 x+16$
• $y^2=22 x^6+x^5+37 x^4+3 x^3+2 x^2+38 x+50$
• $y^2=44 x^6+2 x^5+21 x^4+6 x^3+4 x^2+23 x+47$
• $y^2=34 x^6+8 x^5+3 x^4+44 x^3+50 x^2+9 x+44$
• $y^2=15 x^6+16 x^5+6 x^4+35 x^3+47 x^2+18 x+35$
• $y^2=34 x^6+43 x^5+50 x^4+50 x^2+10 x+34$
• $y^2=15 x^6+33 x^5+47 x^4+47 x^2+20 x+15$
• $y^2=46 x^6+5 x^5+50 x^4+12 x^3+x^2+27$
• $y^2=39 x^6+10 x^5+47 x^4+24 x^3+2 x^2+1$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53^{4}}$.

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{106})\).

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

The base change of $A$ to $\F_{53^{4}}$ is 1.7890481.iic^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $53$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{53^{2}}$

  The base change of $A$ to $\F_{53^{2}}$ is the simple isogeny class 2.2809.a_iic and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{-1}) \) with the following ramification data at primes above $53$, and unramified at all archimedean places:

  $v$	($ 53 $,\( \pi + 23 \))	($ 53 $,\( \pi + 30 \))
  $\operatorname{inv}_v$	$1/2$	$1/2$

  where $\pi$ is a root of $x^{2} + 1$.

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_aec	$8$	(not in LMFDB)
2.53.a_ec	$8$	(not in LMFDB)
2.53.a_acb	$24$	(not in LMFDB)