Properties

Label 1.19.a
Base Field $\F_{19}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{19}$
Dimension:  $1$
L-polynomial:  $1 + 19 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-19}) \)
Galois group:  $C_2$
Jacobians:  4

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 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 20 400 6860 129600 2476100 47059600 893871740 16983302400 322687697780 6131071210000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 20 400 6860 129600 2476100 47059600 893871740 16983302400 322687697780 6131071210000

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-19}) \).
Endomorphism algebra over $\overline{\F}_{19}$
The base change of $A$ to $\F_{19^{2}}$ is the simple isogeny class 1.361.bm and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $19$ and $\infty$.
All geometric endomorphisms are defined over $\F_{19^{2}}$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.