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

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

Invariants

Base field:	$\F_{59}$
Dimension:	$2$
L-polynomial:	$1 + 2 x - 55 x^{2} + 118 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.208225049398$, $\pm0.874891716065$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-58})\)
Galois group:	$C_2^2$
Jacobians:	$12$

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3547$	$11729929$	$42323187076$	$146903578058425$	$511139975811910027$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$62$	$3368$	$206072$	$12123396$	$714956782$	$42181115726$	$2488648986298$	$146830449778756$	$8662995475131608$	$511116752925873128$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=26 x^6+17 x^5+22 x^4+47 x^3+35 x^2+11 x+56$
• $y^2=42 x^6+56 x^5+34 x^4+15 x^3+33 x^2+12 x+8$
• $y^2=52 x^6+5 x^5+19 x^4+33 x^3+53 x^2+53 x+12$
• $y^2=14 x^6+47 x^5+6 x^4+22 x^3+13 x^2+17 x+42$
• $y^2=2 x^6+53 x^5+16 x^4+49 x^3+26 x^2+26 x+7$
• $y^2=2 x^6+6 x^5+25 x^4+33 x^3+25 x^2+4 x+3$
• $y^2=58 x^6+10 x^5+x^4+15 x^3+47 x^2+14 x+42$
• $y^2=27 x^6+54 x^5+22 x^4+54 x^3+6 x^2+10 x+29$
• $y^2=20 x^6+26 x^5+35 x^4+53 x^3+32 x^2+11 x+47$
• $y^2=17 x^6+5 x^5+45 x^4+3 x^3+15 x^2+26 x+51$
• $y^2=39 x^6+23 x^5+50 x^4+51 x^3+56 x^2+41 x+23$
• $y^2=36 x^6+7 x^5+2 x^4+41 x^3+20 x^2+49 x+54$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{59^{3}}$.

Endomorphism algebra over $\F_{59}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-58})\).

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

The base change of $A$ to $\F_{59^{3}}$ is 1.205379.ni^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-58}) \)$)$

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.ac_acd	$2$	(not in LMFDB)
2.59.ae_es	$3$	(not in LMFDB)
2.59.ac_acd	$6$	(not in LMFDB)