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

• Label: 2.47.a_ada
• 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:	$1 - 78 x^{2} + 2209 x^{4}$
Frobenius angles:	$\pm0.0942308666757$, $\pm0.905769133324$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{43})\)
Galois group:	$C_2^2$
Jacobians:	$30$
Isomorphism classes:	34

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2132$	$4545424$	$10779257684$	$23795040096256$	$52599132687006932$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$2054$	$103824$	$4876350$	$229345008$	$10779300038$	$506623120464$	$23811300629374$	$1119130473102768$	$52599133138183814$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{47^{2}}$.

Endomorphism algebra over $\F_{47}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{43})\).

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

The base change of $A$ to $\F_{47^{2}}$ is 1.2209.ada^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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.ai_eg	$4$	(not in LMFDB)
2.47.a_da	$4$	(not in LMFDB)
2.47.i_eg	$4$	(not in LMFDB)