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

• Label: 2.53.ay_jl
• Base field: $\F_{53}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$1 - 24 x + 245 x^{2} - 1272 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.0672921504477$, $\pm0.266041182886$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:	$C_2^2$
Jacobians:	$10$
Isomorphism classes:	10

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1759$	$7653409$	$22164272464$	$62273714959161$	$174887761847933839$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$2724$	$148878$	$7892260$	$418196190$	$22164183798$	$1174708974966$	$62259677791684$	$3299763591802134$	$174887471201742564$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53^{6}}$.

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\).

Endomorphism algebra over $\overline{\F}_{53}$

The base change of $A$ to $\F_{53^{6}}$ is 1.22164361129.afbeg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{53^{2}}$

  The base change of $A$ to $\F_{53^{2}}$ is the simple isogeny class 2.2809.adi_gul and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{5})\).

• Endomorphism algebra over $\F_{53^{3}}$

  The base change of $A$ to $\F_{53^{3}}$ is the simple isogeny class 2.148877.a_afbeg and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{5})\).

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_di	$3$	(not in LMFDB)
2.53.y_jl	$3$	(not in LMFDB)
2.53.a_di	$6$	(not in LMFDB)