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

• Label: 2.53.a_adu
• 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:	$1 - 98 x^{2} + 2809 x^{4}$
Frobenius angles:	$\pm0.0622296275013$, $\pm0.937770372499$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2}, \sqrt{51})\)
Galois group:	$C_2^2$
Jacobians:	$8$
Isomorphism classes:	48

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2712$	$7354944$	$22164245784$	$62196819158016$	$174887470679011032$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$2614$	$148878$	$7882510$	$418195494$	$22164130438$	$1174711139838$	$62259690196894$	$3299763591802134$	$174887470992509014$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

• $y^2=13 x^6+43 x^5+35 x^4+37 x^3+50 x^2+52 x+33$
• $y^2=47 x^6+37 x^5+14 x^4+3 x^3+4 x^2+47 x+12$
• $y^2=4 x^6+12 x^5+24 x^4+16 x^3+22 x^2+31 x+29$
• $y^2=22 x^6+24 x^5+7 x^4+x^3+48 x^2+9 x+49$
• $y^2=42 x^6+18 x^5+38 x^3+39 x^2+16 x+24$
• $y^2=2 x^6+4 x^5+16 x^4+21 x^3+9 x^2+46 x+24$
• $y^2=12 x^6+17 x^5+51 x^4+27 x^3+x^2+18 x+23$
• $y^2=44 x^5+37 x^4+41 x^3+14 x^2+37 x$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53^{2}}$.

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{51})\).

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

The base change of $A$ to $\F_{53^{2}}$ is 1.2809.adu^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-102}) \)$)$

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.a_du	$4$	(not in LMFDB)
2.53.ae_i	$8$	(not in LMFDB)
2.53.e_i	$8$	(not in LMFDB)