Properties

Label 2.23.ap_du
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:  $1 - 15 x + 98 x^{2} - 345 x^{3} + 529 x^{4}$
Frobenius angles:  $\pm0.0252283536038$, $\pm0.308104979730$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{17})\)
Galois group:  $C_2^2$
Jacobians:  $2$
Isomorphism classes:  2

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $268$ $264784$ $148014256$ $78250026816$ $41392253570068$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $9$ $501$ $12168$ $279625$ $6431019$ $147992622$ $3404623413$ $78310472689$ $1801152661464$ $41426514634461$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{23^{6}}$.

Endomorphism algebra over $\F_{23}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\).
Endomorphism algebra over $\overline{\F}_{23}$
The base change of $A$ to $\F_{23^{6}}$ is 1.148035889.abgac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.23.p_du$2$(not in LMFDB)
2.23.a_bd$3$(not in LMFDB)
2.23.p_du$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.23.p_du$2$(not in LMFDB)
2.23.a_bd$3$(not in LMFDB)
2.23.p_du$3$(not in LMFDB)
2.23.a_abd$12$(not in LMFDB)