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

• Label: 2.53.ac_ec
• Base field: $\F_{53}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{53}$
Dimension:	$2$
L-polynomial:	$( 1 - 2 x + 53 x^{2} )( 1 + 53 x^{2} )$
	$1 - 2 x + 106 x^{2} - 106 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.456138099416$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$48$
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2808$	$8491392$	$22210811064$	$62177640505344$	$174876597282669048$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$52$	$3018$	$149188$	$7880078$	$418169492$	$22164860538$	$1174712921252$	$62259667505566$	$3299763483137524$	$174887471362191018$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 48 curves (of which all are hyperelliptic):

• $y^2=21 x^6+33 x^5+24 x^4+47 x^3+40 x^2+21 x+3$
• $y^2=51 x^6+52 x^5+15 x^4+20 x^3+15 x^2+52 x+51$
• $y^2=19 x^6+42 x^5+4 x^4+52 x^3+52 x^2+49 x+32$
• $y^2=19 x^6+51 x^5+43 x^4+x^3+43 x^2+51 x+19$
• $y^2=18 x^6+x^5+50 x^4+40 x^3+34 x^2+46 x+50$
• $y^2=26 x^6+30 x^5+36 x^4+7 x^3+36 x^2+30 x+26$
• $y^2=29 x^6+40 x^5+47 x^4+10 x^3+52 x^2+7 x+47$
• $y^2=27 x^6+15 x^5+48 x^4+51 x^3+48 x^2+15 x+27$
• $y^2=41 x^6+4 x^5+34 x^4+48 x^3+48 x^2+43 x+33$
• $y^2=24 x^6+51 x^5+19 x^3+51 x+24$
• $y^2=43 x^6+46 x^5+22 x^4+9 x^3+41 x^2+15 x+42$
• $y^2=48 x^6+48 x^5+45 x^4+x^3+45 x^2+48 x+48$
• $y^2=34 x^6+15 x^5+19 x^4+5 x^3+33 x^2+36 x+48$
• $y^2=23 x^6+5 x^5+x^4+22 x^3+x^2+5 x+23$
• $y^2=20 x^6+36 x^5+20 x^4+3 x^2+13 x+32$
• $y^2=14 x^6+30 x^5+8 x^4+17 x^3+8 x^2+30 x+14$
• $y^2=12 x^6+10 x^5+49 x^4+33 x^3+38 x^2+28 x+20$
• $y^2=46 x^6+14 x^5+34 x^4+31 x^3+34 x^2+14 x+46$
• $y^2=52 x^6+32 x^5+2 x^4+34 x^3+48 x^2+41 x+9$
• $y^2=29 x^6+6 x^5+49 x^4+4 x^3+49 x^2+6 x+29$
• and

• $y^2=6 x^6+44 x^5+5 x^4+x^3+35 x^2+36 x+44$
• $y^2=22 x^6+21 x^5+35 x^4+48 x^3+35 x^2+21 x+22$
• $y^2=8 x^6+42 x^5+3 x^4+40 x^3+35 x^2+28 x+21$
• $y^2=39 x^6+5 x^5+20 x^4+45 x^3+20 x^2+5 x+39$
• $y^2=4 x^6+23 x^5+11 x^4+2 x^3+44 x^2+50 x+44$
• $y^2=52 x^6+23 x^5+17 x^4+30 x^3+17 x^2+23 x+52$
• $y^2=52 x^6+4 x^5+8 x^4+15 x^3+31 x^2+17 x+15$
• $y^2=x^6+7 x^5+40 x^4+3 x^3+40 x^2+7 x+1$
• $y^2=44 x^6+10 x^5+8 x^4+28 x^3+12 x^2+49 x+16$
• $y^2=14 x^6+31 x^5+42 x^4+35 x^3+42 x^2+31 x+14$
• $y^2=49 x^6+22 x^5+10 x^4+22 x^3+17 x^2+18 x+6$
• $y^2=24 x^6+15 x^5+17 x^4+15 x^3+17 x^2+15 x+24$
• $y^2=28 x^6+46 x^5+35 x^4+51 x^3+45 x^2+49 x+13$
• $y^2=10 x^6+3 x^5+14 x^4+37 x^3+14 x^2+3 x+10$
• $y^2=7 x^6+x^5+38 x^4+4 x^3+10 x^2+24 x+47$
• $y^2=28 x^6+33 x^5+34 x^4+37 x^3+34 x^2+33 x+28$
• $y^2=34 x^6+6 x^5+21 x^4+23 x^3+8 x^2+7 x+21$
• $y^2=23 x^6+11 x^5+46 x^4+25 x^3+46 x^2+11 x+23$
• $y^2=48 x^6+29 x^5+29 x^4+49 x^3+28 x^2+38 x+18$
• $y^2=38 x^6+13 x^5+12 x^4+30 x^3+12 x^2+13 x+38$
• $y^2=33 x^6+47 x^5+35 x^4+4 x^3+35 x^2+47 x+33$
• $y^2=9 x^6+3 x^5+30 x^4+27 x^3+30 x^2+3 x+9$
• $y^2=23 x^6+29 x^5+28 x^4+36 x^3+40 x^2+17 x+19$
• $y^2=28 x^6+40 x^5+27 x^4+19 x^3+27 x^2+40 x+28$
• $y^2=33 x^6+x^5+47 x^4+42 x^3+x^2+28 x+5$
• $y^2=22 x^6+40 x^5+13 x^4+7 x^3+13 x^2+40 x+22$
• $y^2=17 x^6+5 x^5+34 x^4+13 x^3+19 x^2+5 x+36$
• $y^2=7 x^6+19 x^5+9 x^4+3 x^3+9 x^2+19 x+7$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.ac $\times$ 1.53.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.53.ac : \(\Q(\sqrt{-13}) \).
• 1.53.a : \(\Q(\sqrt{-53}) \).

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

The base change of $A$ to $\F_{53^{2}}$ is 1.2809.dy $\times$ 1.2809.ec. The endomorphism algebra for each factor is:

• 1.2809.dy : \(\Q(\sqrt{-13}) \).
• 1.2809.ec : the quaternion algebra over \(\Q\) ramified at $53$ and $\infty$.

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.c_ec	$2$	(not in LMFDB)