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

• Label: 2.47.a_add
• 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:	$1 - 81 x^{2} + 2209 x^{4}$
Frobenius angles:	$\pm0.0846993382161$, $\pm0.915300661784$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{7}, \sqrt{-13})\)
Galois group:	$C_2^2$
Jacobians:	$5$
Isomorphism classes:	18
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2129$	$4532641$	$10779220676$	$23790386696521$	$52599132642540689$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$2048$	$103824$	$4875396$	$229345008$	$10779226022$	$506623120464$	$23811296995588$	$1119130473102768$	$52599133049251328$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):

• $y^2=22 x^6+12 x^5+11 x^4+34 x^3+40 x^2+44 x+8$
• $y^2=22 x^6+42 x^5+37 x^4+40 x^3+32 x^2+24 x+26$
• $y^2=15 x^6+25 x^5+29 x^4+12 x^3+13 x^2+7 x+24$
• $y^2=13 x^6+26 x^5+17 x^4+46 x^3+40 x^2+39 x+1$
• $y^2=14 x^6+34 x^5+27 x^4+13 x^3+26 x^2+43 x+46$

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 \(\Q(\sqrt{7}, \sqrt{-13})\).

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

The base change of $A$ to $\F_{47^{2}}$ is 1.2209.add^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-91}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.47.a_dd	$4$	(not in LMFDB)