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

• Label: 2.53.ai_eq
• 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:	$1 - 8 x + 120 x^{2} - 424 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.378723670807$, $\pm0.443169091899$
Angle rank:	$2$ (numerical)
Number field:	4.0.2400512.2
Galois group:	$D_{4}$
Jacobians:	$16$
Isomorphism classes:	16
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2498$	$8398276$	$22328105714$	$62224137015056$	$174857994668552098$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$46$	$2986$	$149974$	$7885974$	$418125006$	$22164308506$	$1174714187350$	$62259703882398$	$3299763509557198$	$174887469530995146$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

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

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 4.0.2400512.2.

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.i_eq	$2$	(not in LMFDB)