Properties

Label 1.269.a
Base Field $\F_{269}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{269}$
Dimension:  $1$
L-polynomial:  $1 + 269 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-269}) \)
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
$A(\F_{q^r})$ 270 72900 19465110 5235969600 1408514752350 378890507312100 101921535994725990 27416893172109062400 7375144266114367290030 1983913807587581830522500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 270 72900 19465110 5235969600 1408514752350 378890507312100 101921535994725990 27416893172109062400 7375144266114367290030 1983913807587581830522500

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{269}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-269}) \).
Endomorphism algebra over $\overline{\F}_{269}$
The base change of $A$ to $\F_{269^{2}}$ is the simple isogeny class 1.72361.us and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $269$ and $\infty$.
All geometric endomorphisms are defined over $\F_{269^{2}}$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.