Abelian variety isogeny class 2.43.a_di over $\F_{43}$

• Label: 2.43.a_di
• Base field: $\F_{43}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{43}$
Dimension:	$2$
L-polynomial:	$( 1 + 43 x^{2} )^{2}$
	$1 + 86 x^{2} + 1849 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.5$
Angle rank:	$0$ (numerical)
Jacobians:	$27$
Cyclic group of points:	no
Non-cyclic primes:	$2, 11$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1936$	$3748096$	$6321522064$	$11662935330816$	$21611482607301136$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$44$	$2022$	$79508$	$3411406$	$147008444$	$6321681078$	$271818611108$	$11688186602398$	$502592611936844$	$21611482901318022$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 27 curves (of which all are hyperelliptic):

• $y^2=11 x^6+20 x^5+31 x^4+7 x^2+35 x+39$
• $y^2=5 x^6+4 x^5+25 x^4+13 x^3+27 x^2+9 x+10$
• $y^2=37 x^5+10 x^4+23 x^3+8 x^2+34 x$
• $y^2=36 x^6+11 x^5+30 x+26$
• $y^2=23 x^6+9 x^5+27 x^4+14 x^3+37 x^2+29 x+31$
• $y^2=7 x^6+8 x^5+38 x^4+4 x^3+25 x^2+28 x+28$
• $y^2=21 x^6+24 x^5+28 x^4+12 x^3+32 x^2+41 x+41$
• $y^2=23 x^6+38 x^5+20 x^4+21 x^3+20 x^2+38 x+23$
• $y^2=26 x^6+28 x^5+17 x^4+20 x^3+17 x^2+28 x+26$
• $y^2=28 x^6+2 x^5+22 x^4+29 x^3+6 x^2+29 x+13$
• $y^2=41 x^6+6 x^5+23 x^4+x^3+18 x^2+x+39$
• $y^2=38 x^6+5 x^5+22 x^4+42 x^3+37 x^2+32 x+30$
• $y^2=28 x^6+15 x^5+23 x^4+40 x^3+25 x^2+10 x+4$
• $y^2=35 x^5+10 x^4+33 x^3+32 x^2+23 x$
• $y^2=x^6+23 x^5+22 x^4+18 x^3+17 x^2+14 x+2$
• $y^2=42 x^6+25 x^5+2 x^3+8 x^2+25 x+42$
• $y^2=40 x^6+32 x^5+6 x^3+24 x^2+32 x+40$
• $y^2=20 x^6+33 x^5+19 x^4+39 x^3+x^2+19 x+11$
• $y^2=17 x^6+13 x^5+14 x^4+31 x^3+3 x^2+14 x+33$
• $y^2=41 x^6+26 x^5+27 x^4+39 x^3+6 x^2+29 x+19$
• $y^2=37 x^6+35 x^5+38 x^4+31 x^3+18 x^2+x+14$
• $y^2=33 x^6+15 x^4+15 x^2+33$
• $y^2=13 x^6+2 x^4+2 x^2+13$
• $y^2=33 x^6+2 x^4+6 x^2+31$
• $y^2=12 x^6+16 x^4+5 x^2+23$
• $y^2=39 x^6+20 x^5+x^4+6 x^3+33 x^2+20 x+1$
• $y^2=31 x^6+17 x^5+3 x^4+18 x^3+13 x^2+17 x+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{43^{2}}$.

Endomorphism algebra over $\F_{43}$

The isogeny class factors as 1.43.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

Endomorphism algebra over $\overline{\F}_{43}$

The base change of $A$ to $\F_{43^{2}}$ is 1.1849.di^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $43$ 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
2.43.a_abr	$3$	(not in LMFDB)
2.43.a_adi	$4$	(not in LMFDB)
2.43.a_abr	$6$	(not in LMFDB)