Properties

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

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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

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 contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon 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)