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

• Label: 2.53.ae_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:	$\pm0.456138099416$, $\pm0.456138099416$
Angle rank:	$1$ (numerical)
Jacobians:	$56$
Cyclic group of points:	no
Non-cyclic primes:	$2, 13$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2704$	$8479744$	$22257059344$	$62184201404416$	$174865724039538064$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$50$	$3014$	$149498$	$7880910$	$418143490$	$22164764438$	$1174714702666$	$62259676161694$	$3299763374472914$	$174887470686087014$

Jacobians and polarizations

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

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.ac^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.a_dy	$2$	(not in LMFDB)
2.53.e_eg	$2$	(not in LMFDB)
2.53.c_abx	$3$	(not in LMFDB)