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

• Label: 2.53.e_eg
• 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:	$( 1 + 2 x + 53 x^{2} )^{2}$
	$1 + 4 x + 110 x^{2} + 212 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.543861900584$, $\pm0.543861900584$
Angle rank:	$1$ (numerical)
Jacobians:	$56$
Cyclic group of points:	no
Non-cyclic primes:	$2, 7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3136$	$8479744$	$22072450624$	$62184201404416$	$174909219716478016$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$58$	$3014$	$148258$	$7880910$	$418247498$	$22164764438$	$1174707577010$	$62259676161694$	$3299763809131354$	$174887470686087014$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 56 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.c^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$

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.ae_eg	$2$	(not in LMFDB)
2.53.a_dy	$2$	(not in LMFDB)
2.53.ac_abx	$3$	(not in LMFDB)