Properties

Label 1.389.a
Base Field $\F_{389}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{389}$
Dimension:  $1$
L-polynomial:  $1 + 389 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-389}) \)
Galois group:  $C_2$
Jacobians:  22

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

This isogeny class contains the Jacobians of 22 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 390 152100 58863870 22897742400 8907339520950 3464955191376900 1347867523649523630 524320466653868601600 203960661546169565063910 79340697341477775488902500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 390 152100 58863870 22897742400 8907339520950 3464955191376900 1347867523649523630 524320466653868601600 203960661546169565063910 79340697341477775488902500

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{389}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-389}) \).
Endomorphism algebra over $\overline{\F}_{389}$
The base change of $A$ to $\F_{389^{2}}$ is the simple isogeny class 1.151321.bdy and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $389$ and $\infty$.
All geometric endomorphisms are defined over $\F_{389^{2}}$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.