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

• Label: 2.47.d_abm
• 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:	$1 + 3 x - 38 x^{2} + 141 x^{3} + 2209 x^{4}$
Frobenius angles:	$\pm0.236880074768$, $\pm0.903546741435$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-179})\)
Galois group:	$C_2^2$
Jacobians:	$35$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2316$	$4696848$	$10861808400$	$23824986928704$	$52605332620869396$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$51$	$2125$	$104616$	$4882489$	$229372041$	$10779316990$	$506621697423$	$23811284781649$	$1119130350617592$	$52599132507923125$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 35 curves (of which all are hyperelliptic):

• $y^2=37 x^6+42 x^5+x^3+44 x^2+19 x+27$
• $y^2=14 x^6+7 x^5+6 x^4+5 x^3+20 x^2+8 x$
• $y^2=2 x^6+29 x^5+x^4+41 x^2+28 x+24$
• $y^2=22 x^6+43 x^5+39 x^4+36 x^3+13 x^2+42 x+22$
• $y^2=3 x^6+21 x^5+36 x^4+34 x^3+20 x^2+33 x+33$
• $y^2=32 x^6+16 x^5+14 x^4+x^3+15 x^2+23 x+28$
• $y^2=26 x^6+22 x^5+11 x^4+16 x^3+19 x^2+18 x+42$
• $y^2=20 x^6+12 x^5+41 x^4+44 x^3+36 x^2+30 x+9$
• $y^2=6 x^6+34 x^5+31 x^4+30 x^3+45 x^2+24 x+6$
• $y^2=35 x^6+16 x^5+28 x^4+22 x^3+2 x^2+3 x+35$
• $y^2=x^6+43 x^5+16 x^4+8 x^3+24 x^2+26 x+2$
• $y^2=39 x^6+39 x^5+16 x^4+39 x^3+30 x^2+7 x+39$
• $y^2=21 x^6+26 x^5+42 x^4+23 x^3+39 x^2+6 x+21$
• $y^2=34 x^6+4 x^5+2 x^4+35 x^3+23 x^2+2 x+9$
• $y^2=20 x^6+43 x^5+33 x^4+28 x^3+15 x^2+45 x+10$
• $y^2=10 x^6+27 x^5+40 x^4+27 x^3+25 x+13$
• $y^2=46 x^5+14 x^4+36 x^3+2 x^2+15 x+19$
• $y^2=38 x^5+6 x^4+6 x^3+44 x^2+x+37$
• $y^2=38 x^6+30 x^5+25 x^4+19 x^3+42 x^2+32 x+40$
• $y^2=7 x^6+10 x^5+34 x^4+11 x^3+22 x^2+9 x+1$
• and

• $y^2=40 x^6+20 x^5+2 x^4+41 x^3+22 x^2+10 x+40$
• $y^2=41 x^6+25 x^5+8 x^4+18 x^3+41 x^2+38 x+32$
• $y^2=41 x^6+38 x^5+7 x^4+46 x^3+9 x^2+20 x+41$
• $y^2=11 x^6+26 x^5+40 x^4+35 x^3+34 x^2+36 x+5$
• $y^2=23 x^6+28 x^5+10 x^4+16 x^3+27 x^2+16 x+23$
• $y^2=x^6+37 x^5+33 x^4+12 x^3+2 x^2+x+16$
• $y^2=45 x^6+44 x^5+18 x^4+x^3+22 x^2+5 x+12$
• $y^2=31 x^6+41 x^5+17 x^4+28 x^2+6 x+18$
• $y^2=42 x^6+13 x^5+7 x^4+2 x^3+11 x^2+20 x+40$
• $y^2=19 x^6+10 x^5+44 x^4+8 x^3+21 x^2+x+30$
• $y^2=8 x^6+30 x^5+38 x^4+28 x^3+45 x^2+44 x+30$
• $y^2=9 x^6+3 x^5+45 x^4+43 x^3+33 x^2+28 x+29$
• $y^2=24 x^6+6 x^5+41 x^4+39 x^3+41 x^2+45 x+24$
• $y^2=34 x^6+23 x^5+14 x^4+3 x^3+21 x^2+24 x+16$
• $y^2=18 x^6+35 x^5+25 x^4+8 x^3+x^2+39 x+15$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{47^{3}}$.

Endomorphism algebra over $\F_{47}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-179})\).

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

The base change of $A$ to $\F_{47^{3}}$ is 1.103823.pg^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-179}) \)$)$

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.ad_abm	$2$	(not in LMFDB)
2.47.ag_dz	$3$	(not in LMFDB)
2.47.ad_abm	$6$	(not in LMFDB)