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

• Label: 2.53.m_fi
• Base field: $\F_{53}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• 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:	$( 1 + 4 x + 53 x^{2} )( 1 + 8 x + 53 x^{2} )$
	$1 + 12 x + 138 x^{2} + 636 x^{3} + 2809 x^{4}$
Frobenius angles:	$\pm0.588585532783$, $\pm0.685159765542$
Angle rank:	$2$ (numerical)
Jacobians:	$36$
Isomorphism classes:	92
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3596$	$8270800$	$21966788108$	$62270522368000$	$174908252552617196$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$66$	$2942$	$147546$	$7891854$	$418245186$	$22164051854$	$1174710862650$	$62259700959646$	$3299763560589378$	$174887470329450782$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 36 curves (of which all are hyperelliptic):

• $y^2=45 x^6+8 x^5+52 x^4+13 x^3+12 x^2+5 x+6$
• $y^2=16 x^6+48 x^5+31 x^4+27 x^3+45 x^2+30 x+47$
• $y^2=19 x^6+43 x^5+37 x^4+5 x^2+28 x+13$
• $y^2=20 x^6+41 x^5+41 x^4+49 x^3+41 x^2+41 x+20$
• $y^2=2 x^6+5 x^5+49 x^4+29 x^3+49 x^2+5 x+2$
• $y^2=31 x^6+36 x^5+31 x^4+38 x^3+27 x^2+29 x+13$
• $y^2=44 x^6+23 x^5+2 x^4+23 x^3+24 x^2+52 x+29$
• $y^2=23 x^6+33 x^5+22 x^4+31 x^3+31 x^2+33 x+30$
• $y^2=20 x^6+38 x^5+21 x^4+31 x^3+21 x^2+38 x+20$
• $y^2=37 x^6+18 x^5+45 x^4+36 x^3+39 x^2+22 x+7$
• $y^2=9 x^6+51 x^5+26 x^4+28 x^3+19 x^2+27 x+6$
• $y^2=7 x^6+3 x^5+33 x^4+40 x^3+33 x^2+3 x+7$
• $y^2=40 x^6+35 x^5+47 x^4+26 x^3+47 x^2+35 x+40$
• $y^2=8 x^6+x^5+37 x^4+51 x^3+45 x+8$
• $y^2=48 x^6+6 x^5+47 x^4+36 x^3+47 x^2+6 x+48$
• $y^2=18 x^6+21 x^5+11 x^4+31 x^3+34 x^2+39 x+22$
• $y^2=34 x^6+45 x^4+27 x^3+22 x^2+20$
• $y^2=34 x^6+11 x^5+7 x^4+30 x^3+36 x^2+3 x+29$
• $y^2=49 x^6+27 x^5+47 x^4+38 x^3+36 x^2+14 x+44$
• $y^2=43 x^6+19 x^5+4 x^4+37 x^3+4 x^2+19 x+43$
• and

• $y^2=24 x^6+50 x^5+49 x^4+33 x^3+49 x^2+50 x+24$
• $y^2=30 x^6+11 x^5+21 x^4+40 x^3+21 x^2+11 x+30$
• $y^2=37 x^6+30 x^5+29 x^4+21 x^3+29 x^2+30 x+37$
• $y^2=34 x^5+20 x^4+9 x^3+14 x^2+3$
• $y^2=17 x^6+20 x^5+3 x^4+10 x^3+x^2+50 x+41$
• $y^2=22 x^6+51 x^5+15 x^4+44 x^3+32 x^2+40 x+14$
• $y^2=44 x^6+41 x^5+15 x^4+47 x^3+33 x^2+21 x+11$
• $y^2=50 x^6+44 x^5+18 x^4+34 x^3+45 x^2+11 x+11$
• $y^2=48 x^6+45 x^5+43 x^4+10 x^3+17 x^2+51 x+52$
• $y^2=15 x^6+19 x^5+30 x^4+4 x^3+45 x^2+3 x+44$
• $y^2=17 x^6+7 x^5+27 x^4+43 x^3+12 x^2+3 x+22$
• $y^2=19 x^6+10 x^5+10 x^4+45 x^3+10 x^2+10 x+19$
• $y^2=37 x^6+30 x^5+24 x^4+19 x^3+29 x^2+30 x+16$
• $y^2=40 x^6+23 x^5+19 x^4+30 x^3+10 x^2+51 x+14$
• $y^2=48 x^6+25 x^5+35 x^4+15 x^3+7 x^2+38 x+29$
• $y^2=41 x^6+11 x^5+47 x^4+18 x^3+5 x^2+32 x+36$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{53}$

The isogeny class factors as 1.53.e $\times$ 1.53.i 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.e : \(\Q(\sqrt{-1}) \).
• 1.53.i : \(\Q(\sqrt{-37}) \).

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.am_fi	$2$	(not in LMFDB)
2.53.ae_cw	$2$	(not in LMFDB)
2.53.e_cw	$2$	(not in LMFDB)