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

• Label: 2.53.ao_du
• 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:	$1 - 14 x + 98 x^{2} - 742 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.0120231262017$, $\pm0.512023126202$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{57})\)
Galois group:	$C_2^2$
Jacobians:	$8$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2152$	$7884928$	$22037583976$	$62172089565184$	$174873882049061032$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$40$	$2810$	$148024$	$7879374$	$418163000$	$22164361130$	$1174708481000$	$62259660279454$	$3299763521797192$	$174887470365513050$

Jacobians and polarizations

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

• $y^2=9 x^6+41 x^5+23 x^4+24 x^3+11 x^2+23 x+6$
• $y^2=34 x^6+18 x^5+44 x^4+20 x^3+40 x^2+51 x+32$
• $y^2=9 x^6+4 x^5+16 x^4+16 x^2+49 x+9$
• $y^2=35 x^6+48 x^5+9 x^4+4 x^3+22 x^2+49 x+14$
• $y^2=46 x^6+19 x^5+28 x^4+38 x^3+8 x^2+3 x+31$
• $y^2=18 x^6+34 x^5+24 x^4+24 x^2+19 x+18$
• $y^2=15 x^6+36 x^5+33 x^4+23 x^3+38 x^2+25 x+32$
• $y^2=49 x^6+6 x^5+4 x^4+26 x^3+36 x^2+16 x+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53^{4}}$.

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{57})\).

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

The base change of $A$ to $\F_{53^{4}}$ is 1.7890481.aifq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-57}) \)$)$

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.a_aifq and its endomorphism algebra is \(\Q(i, \sqrt{57})\).

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.o_du	$2$	(not in LMFDB)
2.53.a_ai	$8$	(not in LMFDB)
2.53.a_i	$8$	(not in LMFDB)