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

• Label: 2.53.e_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:	$\pm0.282846464884$, $\pm0.860301140065$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-298 -46 \sqrt{5}})\)
Galois group:	$D_{4}$
Jacobians:	$60$
Isomorphism classes:	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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3011$	$7765369$	$22295804624$	$62308458900041$	$174882036073234891$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$58$	$2764$	$149758$	$7896660$	$418182498$	$22164448438$	$1174706814010$	$62259694425444$	$3299763528854854$	$174887471349340764$

Jacobians and polarizations

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

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

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

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.ae_ap	$2$	(not in LMFDB)