Properties

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

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:

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

 $r$ 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.