Abelian variety isogeny class 2.23.am_cu over $\F_{23}$

• Label: 2.23.am_cu
• Base field: $\F_{23}$
• 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:	$\F_{23}$
Dimension:	$2$
L-polynomial:	$1 - 12 x + 72 x^{2} - 276 x^{3} + 529 x^{4}$
Frobenius angles:	$\pm0.0956038575290$, $\pm0.404396142471$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{10})\)
Galois group:	$C_2^2$
Jacobians:	$7$
Isomorphism classes:	14
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$314$	$279460$	$148474586$	$78097891600$	$41391692364794$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$530$	$12204$	$279078$	$6430932$	$148035890$	$3404985204$	$78311812798$	$1801153952172$	$41426511213650$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):

• $y^2=13 x^6+5 x^5+21 x^4+21 x^2+18 x+13$
• $y^2=10 x^6+13 x^5+19 x^4+5 x^3+19 x^2+11 x+7$
• $y^2=9 x^6+14 x^5+13 x^4+15 x^3+9 x^2+16 x+21$
• $y^2=4 x^6+17 x^5+17 x^4+7 x^2+12 x+9$
• $y^2=9 x^6+6 x^5+20 x^4+16 x^3+18 x^2+18 x+14$
• $y^2=7 x^6+19 x^5+10 x^4+6 x^3+10 x^2+9 x+18$
• $y^2=16 x^6+17 x^5+20 x^3+16 x^2+6 x+5$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{23^{4}}$.

Endomorphism algebra over $\F_{23}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{10})\).

Endomorphism algebra over $\overline{\F}_{23}$

The base change of $A$ to $\F_{23^{4}}$ is 1.279841.aos^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{23^{2}}$

  The base change of $A$ to $\F_{23^{2}}$ is the simple isogeny class 2.529.a_aos and its endomorphism algebra is \(\Q(i, \sqrt{10})\).

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
2.23.m_cu	$2$	(not in LMFDB)
2.23.a_aba	$8$	(not in LMFDB)
2.23.a_ba	$8$	(not in LMFDB)