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

• Label: 2.53.c_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:	$( 1 + 53 x^{2} )( 1 + 2 x + 53 x^{2} )$
	$1 + 2 x + 106 x^{2} + 106 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.543861900584$
Angle rank:	$1$ (numerical)
Jacobians:	$48$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3024$	$8491392$	$22118506704$	$62177640505344$	$174898345121139024$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$3018$	$148568$	$7880078$	$418221496$	$22164860538$	$1174709358424$	$62259667505566$	$3299763700466744$	$174887471362191018$

Jacobians and polarizations

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

• $y^2=37 x^6+43 x^5+12 x^4+50 x^3+20 x^2+37 x+28$
• $y^2=49 x^6+51 x^5+30 x^4+40 x^3+30 x^2+51 x+49$
• $y^2=36 x^6+21 x^5+2 x^4+26 x^3+26 x^2+51 x+16$
• $y^2=38 x^6+49 x^5+33 x^4+2 x^3+33 x^2+49 x+38$
• $y^2=36 x^6+2 x^5+47 x^4+27 x^3+15 x^2+39 x+47$
• $y^2=13 x^6+15 x^5+18 x^4+30 x^3+18 x^2+15 x+13$
• $y^2=5 x^6+27 x^5+41 x^4+20 x^3+51 x^2+14 x+41$
• $y^2=40 x^6+34 x^5+24 x^4+52 x^3+24 x^2+34 x+40$
• $y^2=47 x^6+2 x^5+17 x^4+24 x^3+24 x^2+48 x+43$
• $y^2=48 x^6+49 x^5+38 x^3+49 x+48$
• $y^2=48 x^6+23 x^5+11 x^4+31 x^3+47 x^2+34 x+21$
• $y^2=24 x^6+24 x^5+49 x^4+27 x^3+49 x^2+24 x+24$
• $y^2=15 x^6+30 x^5+38 x^4+10 x^3+13 x^2+19 x+43$
• $y^2=38 x^6+29 x^5+27 x^4+11 x^3+27 x^2+29 x+38$
• $y^2=10 x^6+18 x^5+10 x^4+28 x^2+33 x+16$
• $y^2=28 x^6+7 x^5+16 x^4+34 x^3+16 x^2+7 x+28$
• $y^2=6 x^6+5 x^5+51 x^4+43 x^3+19 x^2+14 x+10$
• $y^2=39 x^6+28 x^5+15 x^4+9 x^3+15 x^2+28 x+39$
• $y^2=26 x^6+16 x^5+x^4+17 x^3+24 x^2+47 x+31$
• $y^2=5 x^6+12 x^5+45 x^4+8 x^3+45 x^2+12 x+5$
• and

• $y^2=12 x^6+35 x^5+10 x^4+2 x^3+17 x^2+19 x+35$
• $y^2=11 x^6+37 x^5+44 x^4+24 x^3+44 x^2+37 x+11$
• $y^2=4 x^6+21 x^5+28 x^4+20 x^3+44 x^2+14 x+37$
• $y^2=46 x^6+29 x^5+10 x^4+49 x^3+10 x^2+29 x+46$
• $y^2=2 x^6+38 x^5+32 x^4+x^3+22 x^2+25 x+22$
• $y^2=26 x^6+38 x^5+35 x^4+15 x^3+35 x^2+38 x+26$
• $y^2=26 x^6+2 x^5+4 x^4+34 x^3+42 x^2+35 x+34$
• $y^2=2 x^6+14 x^5+27 x^4+6 x^3+27 x^2+14 x+2$
• $y^2=22 x^6+5 x^5+4 x^4+14 x^3+6 x^2+51 x+8$
• $y^2=7 x^6+42 x^5+21 x^4+44 x^3+21 x^2+42 x+7$
• $y^2=51 x^6+11 x^5+5 x^4+11 x^3+35 x^2+9 x+3$
• $y^2=12 x^6+34 x^5+35 x^4+34 x^3+35 x^2+34 x+12$
• $y^2=3 x^6+39 x^5+17 x^4+49 x^3+37 x^2+45 x+26$
• $y^2=20 x^6+6 x^5+28 x^4+21 x^3+28 x^2+6 x+20$
• $y^2=30 x^6+27 x^5+19 x^4+2 x^3+5 x^2+12 x+50$
• $y^2=3 x^6+13 x^5+15 x^4+21 x^3+15 x^2+13 x+3$
• $y^2=15 x^6+12 x^5+42 x^4+46 x^3+16 x^2+14 x+42$
• $y^2=38 x^6+32 x^5+23 x^4+39 x^3+23 x^2+32 x+38$
• $y^2=43 x^6+5 x^5+5 x^4+45 x^3+3 x^2+23 x+36$
• $y^2=19 x^6+33 x^5+6 x^4+15 x^3+6 x^2+33 x+19$
• $y^2=43 x^6+50 x^5+44 x^4+2 x^3+44 x^2+50 x+43$
• $y^2=31 x^6+28 x^5+15 x^4+40 x^3+15 x^2+28 x+31$
• $y^2=38 x^6+41 x^5+14 x^4+18 x^3+20 x^2+35 x+36$
• $y^2=14 x^6+20 x^5+40 x^4+36 x^3+40 x^2+20 x+14$
• $y^2=43 x^6+27 x^5+50 x^4+21 x^3+27 x^2+14 x+29$
• $y^2=11 x^6+20 x^5+33 x^4+30 x^3+33 x^2+20 x+11$
• $y^2=35 x^6+29 x^5+17 x^4+33 x^3+36 x^2+29 x+18$
• $y^2=30 x^6+36 x^5+31 x^4+28 x^3+31 x^2+36 x+30$

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.a $\times$ 1.53.c 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.a : \(\Q(\sqrt{-53}) \).
• 1.53.c : \(\Q(\sqrt{-13}) \).

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.ac_ec	$2$	(not in LMFDB)