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

• Label: 2.43.m_cu
• Base field: $\F_{43}$
• 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:	$\F_{43}$
Dimension:	$2$
L-polynomial:	$1 + 12 x + 72 x^{2} + 516 x^{3} + 1849 x^{4}$
Frobenius angles:	$\pm0.473975234504$, $\pm0.973975234504$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{8})\)
Galois group:	$C_2^2$
Jacobians:	$18$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2450$	$3415300$	$6375980450$	$11664274090000$	$21613336463311250$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$56$	$1850$	$80192$	$3411798$	$147021056$	$6321363050$	$271820052392$	$11688189424798$	$502592615078456$	$21611482313284250$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{43^{4}}$.

Endomorphism algebra over $\F_{43}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{8})\).

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

The base change of $A$ to $\F_{43^{4}}$ is 1.3418801.afes^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{43^{2}}$

  The base change of $A$ to $\F_{43^{2}}$ is the simple isogeny class 2.1849.a_afes and its endomorphism algebra is \(\Q(\zeta_{8})\).

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.am_cu	$2$	(not in LMFDB)
2.43.au_he	$8$	(not in LMFDB)
2.43.a_ao	$8$	(not in LMFDB)