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

• Label: 2.59.q_gr
• 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:	$( 1 + 5 x + 59 x^{2} )( 1 + 11 x + 59 x^{2} )$
	$1 + 16 x + 173 x^{2} + 944 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.605523279018$, $\pm0.754046748139$
Angle rank:	$2$ (numerical)
Jacobians:	$66$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4615$	$12437425$	$41898809680$	$146894260137625$	$511126143837689575$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$76$	$3572$	$204004$	$12122628$	$714937436$	$42180398102$	$2488651199684$	$146830434883588$	$8662995993738556$	$511116751710973652$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 66 curves (of which all are hyperelliptic):

• $y^2=48 x^6+41 x^5+3 x^4+4 x^3+2 x^2+43 x+17$
• $y^2=52 x^6+11 x^5+41 x^4+24 x^3+26 x^2+53 x+22$
• $y^2=4 x^6+48 x^5+21 x^4+16 x^3+8 x^2+17 x+58$
• $y^2=21 x^6+51 x^5+39 x^4+21 x^3+22 x^2+45 x+56$
• $y^2=32 x^6+49 x^5+8 x^4+20 x^3+11 x^2+9 x+9$
• $y^2=7 x^6+17 x^5+28 x^4+43 x^3+12 x^2+51 x+6$
• $y^2=5 x^6+24 x^5+46 x^4+55 x^3+49 x^2+54 x+20$
• $y^2=9 x^6+43 x^5+31 x^4+10 x^3+55 x^2+37 x+26$
• $y^2=45 x^6+35 x^5+32 x^4+36 x^3+x^2+33 x+55$
• $y^2=36 x^6+56 x^5+56 x^4+11 x^3+37 x^2+55 x+57$
• $y^2=36 x^6+18 x^5+45 x^4+30 x^3+46 x^2+30 x+51$
• $y^2=35 x^6+49 x^5+14 x^4+51 x^3+26 x^2+56 x+49$
• $y^2=30 x^6+10 x^5+13 x^4+35 x^3+43 x^2+34 x+56$
• $y^2=35 x^6+9 x^5+14 x^4+26 x^3+40 x^2+33 x+49$
• $y^2=29 x^6+3 x^5+56 x^4+49 x^3+12 x^2+17 x+20$
• $y^2=18 x^6+21 x^5+3 x^4+51 x^3+16 x^2+27 x+56$
• $y^2=57 x^6+33 x^5+16 x^4+39 x^3+11 x^2+21 x+27$
• $y^2=11 x^6+25 x^5+x^4+36 x^3+38 x^2+20 x+28$
• $y^2=10 x^6+24 x^5+46 x^4+40 x^3+12 x^2+34 x+28$
• $y^2=34 x^6+27 x^5+27 x^4+31 x^3+4 x^2+41 x+11$
• and

• $y^2=5 x^6+52 x^5+15 x^4+11 x^3+35 x^2+34 x+22$
• $y^2=8 x^6+9 x^5+23 x^4+7 x^3+37 x^2+58 x+12$
• $y^2=36 x^6+4 x^5+33 x^4+48 x^3+48 x^2+9 x+6$
• $y^2=47 x^6+50 x^5+29 x^4+15 x^3+47 x^2+7 x+1$
• $y^2=21 x^6+53 x^5+14 x^4+15 x^3+38 x^2+16 x+25$
• $y^2=15 x^6+42 x^5+52 x^4+43 x^3+18 x^2+8 x+36$
• $y^2=12 x^6+25 x^5+56 x^4+58 x^3+19 x^2+9 x+24$
• $y^2=16 x^6+50 x^5+50 x^4+24 x^3+11 x^2+39 x+17$
• $y^2=51 x^6+38 x^5+25 x^4+16 x^3+16 x^2+47 x+20$
• $y^2=3 x^6+14 x^5+10 x^4+38 x^3+35 x^2+22 x+30$
• $y^2=23 x^6+50 x^5+51 x^4+14 x^3+38 x^2+9 x+45$
• $y^2=44 x^6+11 x^5+16 x^4+28 x^3+19 x^2+54 x+50$
• $y^2=20 x^6+x^5+55 x^4+38 x^3+31 x^2+10 x+6$
• $y^2=44 x^6+35 x^5+50 x^4+57 x^3+19 x^2+47 x+55$
• $y^2=4 x^6+8 x^5+58 x^4+35 x^3+32 x^2+50 x+26$
• $y^2=21 x^6+50 x^5+22 x^4+35 x^3+19 x^2+39 x+26$
• $y^2=10 x^6+15 x^5+31 x^4+6 x^3+40 x^2+27 x+8$
• $y^2=9 x^6+14 x^5+23 x^4+45 x^3+20 x^2+57 x+41$
• $y^2=21 x^6+21 x^5+39 x^4+54 x^3+24 x^2+57 x+7$
• $y^2=43 x^6+34 x^5+40 x^4+x^3+43 x^2+54 x+36$
• $y^2=11 x^6+9 x^5+50 x^4+x^3+40 x^2+3 x+47$
• $y^2=41 x^6+29 x^5+36 x^4+43 x^3+10 x^2+26 x+18$
• $y^2=5 x^6+12 x^5+41 x^3+14 x^2+7 x+1$
• $y^2=21 x^6+23 x^5+28 x^4+58 x^3+3 x^2+22 x+36$
• $y^2=41 x^6+54 x^5+54 x^4+14 x^3+7 x^2+16 x+57$
• $y^2=14 x^6+55 x^5+13 x^4+4 x^3+14 x^2+39 x+30$
• $y^2=28 x^6+7 x^5+38 x^4+35 x^3+51 x^2+56 x+30$
• $y^2=46 x^6+9 x^5+20 x^4+45 x^3+7 x^2+16 x+26$
• $y^2=57 x^6+19 x^5+53 x^4+5 x^3+36 x^2+9 x+11$
• $y^2=36 x^6+30 x^4+55 x^3+37 x^2+27 x+25$
• $y^2=41 x^6+40 x^5+48 x^4+57 x^3+57 x^2+5 x+41$
• $y^2=4 x^6+52 x^5+16 x^4+38 x^3+42 x^2+21 x+35$
• $y^2=44 x^6+39 x^5+34 x^4+57 x^3+44 x^2+55 x+49$
• $y^2=7 x^6+29 x^5+3 x^4+24 x^3+55 x^2+30 x+7$
• $y^2=32 x^6+44 x^5+2 x^4+31 x^3+41 x^2+41 x+35$
• $y^2=12 x^6+12 x^5+46 x^4+50 x^3+29 x^2+12 x+15$
• $y^2=x^6+24 x^5+24 x^4+29 x^3+43 x^2+50 x+15$
• $y^2=25 x^6+44 x^5+28 x^4+8 x^3+7 x^2+47 x+5$
• $y^2=x^6+53 x^5+18 x^4+52 x^3+26 x^2+14 x+18$
• $y^2=27 x^6+17 x^5+17 x^4+46 x^3+41 x^2+31 x+25$
• $y^2=3 x^6+3 x^5+44 x^4+40 x^3+46 x^2+19 x+44$
• $y^2=12 x^6+44 x^5+56 x^4+10 x^3+5 x^2+39 x+19$
• $y^2=28 x^6+25 x^5+22 x^4+55 x^3+26 x^2+35 x+9$
• $y^2=44 x^6+43 x^5+56 x^4+32 x^3+28 x^2+18$
• $y^2=41 x^6+15 x^5+23 x^4+55 x^3+25 x^2+2 x+1$
• $y^2=4 x^6+35 x^5+25 x^4+17 x^3+16 x+53$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The isogeny class factors as 1.59.f $\times$ 1.59.l 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.f : \(\Q(\sqrt{-211}) \).
• 1.59.l : \(\Q(\sqrt{-115}) \).

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
2.59.aq_gr	$2$	(not in LMFDB)
2.59.ag_cl	$2$	(not in LMFDB)
2.59.g_cl	$2$	(not in LMFDB)