Abelian variety isogeny class 2.47.as_gg over $\F_{47}$

• Label: 2.47.as_gg
• Base field: $\F_{47}$
• 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:	$\F_{47}$
Dimension:	$2$
L-polynomial:	$1 - 18 x + 162 x^{2} - 846 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.128711839066$, $\pm0.371288160934$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{13})\)
Galois group:	$C_2^2$
Jacobians:	$26$
Isomorphism classes:	35

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1508$	$4879888$	$10818532244$	$23813306892544$	$52595908202521268$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$2210$	$104202$	$4880094$	$229330950$	$10779215330$	$506624911410$	$23811306095614$	$1119130563832974$	$52599132235830050$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{47^{4}}$.

Endomorphism algebra over $\F_{47}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{13})\).

Endomorphism algebra over $\overline{\F}_{47}$

The base change of $A$ to $\F_{47^{4}}$ is 1.4879681.hy^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{47^{2}}$

  The base change of $A$ to $\F_{47^{2}}$ is the simple isogeny class 2.2209.a_hy and its endomorphism algebra is \(\Q(i, \sqrt{13})\).

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
2.47.s_gg	$2$	(not in LMFDB)
2.47.s_gg	$4$	(not in LMFDB)
2.47.a_acq	$8$	(not in LMFDB)