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

• Label: 2.53.g_el
• 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:	$( 1 + 3 x + 53 x^{2} )^{2}$
	$1 + 6 x + 115 x^{2} + 318 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.566057977562$, $\pm0.566057977562$
Angle rank:	$1$ (numerical)
Jacobians:	$42$
Cyclic group of points:	no
Non-cyclic primes:	$3, 19$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3249$	$8450649$	$22030871184$	$62199894929481$	$174916932642334089$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$3004$	$147978$	$7882900$	$418265940$	$22164551638$	$1174706834676$	$62259693229924$	$3299763811520034$	$174887469556975564$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 42 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.d^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-203}) \)$)$

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.ag_el	$2$	(not in LMFDB)
2.53.a_dt	$2$	(not in LMFDB)
2.53.ad_abs	$3$	(not in LMFDB)