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

• Label: 2.59.ax_jd
• 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:	$1 - 23 x + 237 x^{2} - 1357 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0542088030623$, $\pm0.329038588493$
Angle rank:	$2$ (numerical)
Number field:	4.0.3317429.1
Galois group:	$D_{4}$
Jacobians:	$11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2339$	$11926561$	$42203847077$	$146810968680941$	$511074726590659024$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$3427$	$205495$	$12115755$	$714865512$	$42179909887$	$2488648490197$	$146830442376579$	$8662995996347995$	$511116754364144502$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):

• $y^2=8 x^6+47 x^5+11 x^4+6 x^3+15 x^2+12 x+32$
• $y^2=32 x^6+52 x^5+18 x^4+35 x^3+33 x^2+34 x+56$
• $y^2=50 x^6+27 x^5+26 x^4+2 x^3+52 x^2+36 x+37$
• $y^2=36 x^5+3 x^4+8 x^3+3 x^2+51 x+18$
• $y^2=50 x^6+12 x^5+33 x^4+42 x^3+37 x^2+7 x+58$
• $y^2=7 x^6+56 x^5+50 x^4+15 x^3+56 x^2+52 x+47$
• $y^2=17 x^6+47 x^5+45 x^4+32 x^3+31 x^2+53 x+6$
• $y^2=43 x^6+26 x^5+17 x^4+11 x^3+10 x^2+47 x+54$
• $y^2=37 x^6+6 x^5+16 x^4+37 x^3+53 x^2+35 x+23$
• $y^2=11 x^6+42 x^5+18 x^4+44 x^3+23 x^2+13 x+11$
• $y^2=57 x^6+52 x^5+44 x^4+58 x^3+39 x^2+19 x+11$

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 4.0.3317429.1.

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.x_jd	$2$	(not in LMFDB)