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

• Label: 2.47.a_adq
• Base field: $\F_{47}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{47}$
Dimension:	$2$
L-polynomial:	$( 1 - 47 x^{2} )^{2}$
	$1 - 94 x^{2} + 2209 x^{4}$
Frobenius angles:	$0$, $0$, $1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{47}) \)
Galois group:	$C_2$
Jacobians:	$6$
Cyclic group of points:	no
Non-cyclic primes:	$2, 23$

This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2116$	$4477456$	$10779007684$	$23768199069696$	$52599131777140036$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$48$	$2022$	$103824$	$4870846$	$229345008$	$10778800038$	$506623120464$	$23811267143038$	$1119130473102768$	$52599131318450022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

• $y^2=36 x^6+38 x^5+36 x^4+34 x^2+25 x+37$
• $y^2=x^6+26 x^5+19 x^4+7 x^3+28 x^2+26 x+46$
• $y^2=43 x^6+17 x^5+38 x^4+28 x^3+8 x^2+22 x+15$
• $y^2=27 x^6+33 x^5+25 x^4+x^2+x+8$
• $y^2=x^6+24 x^5+42 x^4+42 x^2+23 x+1$
• $y^2=15 x^6+23 x^5+2 x^4+28 x^2+4 x+35$

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 the quaternion algebra over \(\Q(\sqrt{47}) \) ramified at both real infinite places.

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

The base change of $A$ to $\F_{47^{2}}$ is 1.2209.adq^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $47$ and $\infty$.

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.a_bv	$3$	(not in LMFDB)
2.47.a_dq	$4$	(not in LMFDB)
2.47.a_a	$8$	(not in LMFDB)