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

• Label: 2.59.af_abi
• 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:	$1 - 5 x - 34 x^{2} - 295 x^{3} + 3481 x^{4}$
Frobenius angles:	$\pm0.0611433876487$, $\pm0.727810054315$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-211})\)
Galois group:	$C_2^2$
Jacobians:	$21$
Isomorphism classes:	42

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3148$	$11798704$	$41869344400$	$146850870322624$	$511078667831860228$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$55$	$3389$	$203860$	$12119049$	$714871025$	$42180199958$	$2488653793835$	$146830416215569$	$8662995877231180$	$511116754709018429$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-3}, \sqrt{-211})\).

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

The base change of $A$ to $\F_{59^{3}}$ is 1.205379.abdg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-211}) \)$)$

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.f_abi	$2$	(not in LMFDB)
2.59.k_fn	$3$	(not in LMFDB)
2.59.ak_fn	$6$	(not in LMFDB)