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

• Label: 2.53.a_acr
• 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:	$1 - 69 x^{2} + 2809 x^{4}$
Frobenius angles:	$\pm0.137186756323$, $\pm0.862813243677$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{7}, \sqrt{-37})\)
Galois group:	$C_2^2$
Jacobians:	$10$
Isomorphism classes:	36
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2741$	$7513081$	$22164614084$	$62273231212921$	$174887470693174661$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$54$	$2672$	$148878$	$7892196$	$418195494$	$22164867038$	$1174711139838$	$62259720504388$	$3299763591802134$	$174887471020836272$

Jacobians and polarizations

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

• $y^2=21 x^6+37 x^5+18 x^4+22 x^3+46 x^2+46 x+29$
• $y^2=18 x^6+40 x^5+17 x^4+18 x^2+10 x+11$
• $y^2=29 x^6+26 x^5+2 x^4+15 x^3+17 x^2+50 x+48$
• $y^2=34 x^6+15 x^5+2 x^4+37 x^3+52 x^2+17 x+9$
• $y^2=26 x^6+21 x^5+17 x^4+29 x^3+14 x^2+39 x+46$
• $y^2=11 x^6+11 x^5+7 x^4+41 x^2+28 x+5$
• $y^2=19 x^6+24 x^5+29 x^4+22 x^3+22 x^2+29 x+13$
• $y^2=x^6+37 x^5+4 x^4+43 x^3+2 x^2+49 x+20$
• $y^2=8 x^6+46 x^5+19 x^4+17 x^3+6 x^2+4 x+40$
• $y^2=30 x^6+22 x^5+15 x^4+4 x^3+32 x^2+41 x+47$

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 \(\Q(\sqrt{7}, \sqrt{-37})\).

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

The base change of $A$ to $\F_{53^{2}}$ is 1.2809.acr^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-259}) \)$)$

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_cr	$4$	(not in LMFDB)