Abelian variety isogeny class 2.59.a_cr over $\F_{59}$

• Label: 2.59.a_cr
• Base field: $\F_{59}$
• 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:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$( 1 - 7 x + 59 x^{2} )( 1 + 7 x + 59 x^{2} )$
	$1 + 69 x^{2} + 3481 x^{4}$
Frobenius angles:	$\pm0.349402864844$, $\pm0.650597135156$
Angle rank:	$1$ (numerical)
Jacobians:	$52$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3551$	$12609601$	$42180141584$	$146883807310969$	$511116753327463151$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$60$	$3620$	$205380$	$12121764$	$714924300$	$42179749526$	$2488651484820$	$146830476384964$	$8662995818654940$	$511116753354284900$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 52 curves (of which all are hyperelliptic):

• $y^2=52 x^6+4 x^5+2 x^4+24 x^3+14 x^2+19 x+18$
• $y^2=45 x^6+8 x^5+4 x^4+48 x^3+28 x^2+38 x+36$
• $y^2=44 x^6+35 x^5+18 x^4+49 x^3+20 x^2+32 x+26$
• $y^2=29 x^6+11 x^5+36 x^4+39 x^3+40 x^2+5 x+52$
• $y^2=54 x^6+58 x^5+37 x^4+30 x^3+50 x^2+40 x+53$
• $y^2=49 x^6+57 x^5+15 x^4+x^3+41 x^2+21 x+47$
• $y^2=28 x^6+26 x^5+31 x^4+5 x^3+41 x^2+47 x+47$
• $y^2=56 x^6+52 x^5+3 x^4+10 x^3+23 x^2+35 x+35$
• $y^2=55 x^6+5 x^5+38 x^4+3 x^3+x^2+36 x+57$
• $y^2=12 x^6+37 x^5+10 x^4+41 x^3+55 x^2+25 x+49$
• $y^2=24 x^6+15 x^5+20 x^4+23 x^3+51 x^2+50 x+39$
• $y^2=45 x^6+30 x^5+43 x^4+55 x^2+23 x+1$
• $y^2=31 x^6+x^5+27 x^4+51 x^2+46 x+2$
• $y^2=36 x^6+x^5+47 x^4+x^3+46 x^2+57 x+22$
• $y^2=13 x^6+2 x^5+35 x^4+2 x^3+33 x^2+55 x+44$
• $y^2=38 x^6+40 x^5+31 x^4+27 x^3+3 x^2+42 x+9$
• $y^2=52 x^6+40 x^5+11 x^4+24 x^3+11 x^2+40 x+52$
• $y^2=45 x^6+21 x^5+22 x^4+48 x^3+22 x^2+21 x+45$
• $y^2=38 x^6+45 x^5+30 x^4+12 x^3+21 x^2+25 x+41$
• $y^2=34 x^6+43 x^5+57 x^4+39 x^3+52 x^2+40 x+27$
• and

• $y^2=45 x^6+20 x^5+42 x^4+12 x^3+29 x^2+47 x+49$
• $y^2=31 x^6+40 x^5+25 x^4+24 x^3+58 x^2+35 x+39$
• $y^2=31 x^6+8 x^5+42 x^4+55 x^3+11 x^2+36 x+30$
• $y^2=3 x^6+16 x^5+25 x^4+51 x^3+22 x^2+13 x+1$
• $y^2=46 x^6+7 x^5+51 x^4+41 x^3+48 x+46$
• $y^2=33 x^6+14 x^5+43 x^4+23 x^3+37 x+33$
• $y^2=9 x^6+22 x^5+51 x^4+51 x^3+46 x^2+6 x+4$
• $y^2=18 x^6+44 x^5+43 x^4+43 x^3+33 x^2+12 x+8$
• $y^2=53 x^6+8 x^5+2 x^4+27 x^3+15 x^2+49 x+4$
• $y^2=47 x^6+16 x^5+4 x^4+54 x^3+30 x^2+39 x+8$
• $y^2=10 x^6+31 x^5+5 x^4+4 x^3+17 x^2+16 x+12$
• $y^2=20 x^6+3 x^5+10 x^4+8 x^3+34 x^2+32 x+24$
• $y^2=45 x^6+7 x^5+28 x^4+22 x^3+28 x^2+7 x+45$
• $y^2=31 x^6+14 x^5+56 x^4+44 x^3+56 x^2+14 x+31$
• $y^2=21 x^6+11 x^5+25 x^4+29 x^3+3 x^2+18 x+20$
• $y^2=42 x^6+22 x^5+50 x^4+58 x^3+6 x^2+36 x+40$
• $y^2=20 x^6+10 x^5+19 x^4+6 x^3+42 x^2+31 x+21$
• $y^2=40 x^6+20 x^5+38 x^4+12 x^3+25 x^2+3 x+42$
• $y^2=30 x^6+17 x^5+22 x^4+30 x^3+17 x^2+5 x+41$
• $y^2=x^6+34 x^5+44 x^4+x^3+34 x^2+10 x+23$
• $y^2=20 x^6+16 x^5+43 x^4+14 x^3+54 x^2+20 x+41$
• $y^2=40 x^6+32 x^5+27 x^4+28 x^3+49 x^2+40 x+23$
• $y^2=25 x^6+36 x^5+18 x^4+29 x^3+37 x^2+45 x+39$
• $y^2=50 x^6+13 x^5+36 x^4+58 x^3+15 x^2+31 x+19$
• $y^2=7 x^6+10 x^5+39 x^4+37 x^3+27 x^2+36 x+5$
• $y^2=14 x^6+20 x^5+19 x^4+15 x^3+54 x^2+13 x+10$
• $y^2=38 x^6+50 x^5+30 x^4+x^3+5 x^2+23 x+52$
• $y^2=17 x^6+41 x^5+x^4+2 x^3+10 x^2+46 x+45$
• $y^2=7 x^6+56 x^5+16 x^4+19 x^3+23 x^2+18 x+6$
• $y^2=54 x^6+17 x^5+16 x^4+9 x^3+58 x^2+47 x+17$
• $y^2=49 x^6+34 x^5+32 x^4+18 x^3+57 x^2+35 x+34$
• $y^2=56 x^6+54 x^5+23 x^4+29 x^3+4 x^2+40 x+17$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59^{2}}$.

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.ah $\times$ 1.59.h and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.59.ah : \(\Q(\sqrt{-187}) \).
• 1.59.h : \(\Q(\sqrt{-187}) \).

Endomorphism algebra over $\overline{\F}_{59}$

The base change of $A$ to $\F_{59^{2}}$ is 1.3481.cr^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-187}) \)$)$

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
2.59.ao_gl	$2$	(not in LMFDB)
2.59.o_gl	$2$	(not in LMFDB)
2.59.a_acr	$4$	(not in LMFDB)