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

• Label: 2.59.m_fl
• Base field: $\F_{59}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 + 12 x + 141 x^{2} + 708 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.549816590759$, $\pm0.715010389924$
Angle rank:	$2$ (numerical)
Number field:	4.0.5593393.1
Galois group:	$D_{4}$
Jacobians:	$40$
Isomorphism classes:	40

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4343$	$12607729$	$41929893056$	$146838399521113$	$511134778930931063$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$72$	$3620$	$204156$	$12118020$	$714949512$	$42180524534$	$2488651834104$	$146830414556548$	$8662995964664676$	$511116754582672580$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 40 curves (of which all are hyperelliptic):

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is 4.0.5593393.1.

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.am_fl	$2$	(not in LMFDB)