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

• Label: 2.53.ae_ap
• 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:	$1 - 4 x - 15 x^{2} - 212 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.139698859935$, $\pm0.717153535116$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-298 -46 \sqrt{5}})\)
Galois group:	$D_{4}$
Jacobians:	$60$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2579$	$7765369$	$22033779344$	$62308458900041$	$174892905810514939$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$50$	$2764$	$147998$	$7896660$	$418208490$	$22164448438$	$1174715465666$	$62259694425444$	$3299763654749414$	$174887471349340764$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 60 curves (of which all are hyperelliptic):

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

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

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 \(\Q(\sqrt{-298 -46 \sqrt{5}})\).

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.e_ap	$2$	(not in LMFDB)