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

• Label: 2.53.g_ar
• 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:	$1 + 6 x - 17 x^{2} + 318 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.301864837093$, $\pm0.968531503760$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-11})\)
Galois group:	$C_2^2$
Jacobians:	$10$
Isomorphism classes:	58

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3117$	$7695873$	$22384947456$	$62254017670329$	$174902026871139357$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$2740$	$150354$	$7889764$	$418230300$	$22163867350$	$1174710776460$	$62259675145924$	$3299763736469322$	$174887470740579700$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{53^{3}}$.

Endomorphism algebra over $\F_{53}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\).

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

The base change of $A$ to $\F_{53^{3}}$ is 1.148877.bck^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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.ag_ar	$2$	(not in LMFDB)
2.53.am_fm	$3$	(not in LMFDB)
2.53.a_cs	$6$	(not in LMFDB)