Properties

Label 1.181.a
Base field $\F_{181}$
Dimension $1$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{181}$
Dimension:  $1$
L-polynomial:  $1 + 181 x^{2}$
Frobenius angles:  $\pm0.5$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-181}) \)
Galois group:  $C_2$
Jacobians:  $10$

This isogeny class is simple and geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $182$ $33124$ $5929742$ $1073217600$ $194264244902$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $182$ $33124$ $5929742$ $1073217600$ $194264244902$ $35161840186564$ $6364290927201662$ $1151936655676934400$ $208500535066053616022$ $37738596847344232989604$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{181^{2}}$.

Endomorphism algebra over $\F_{181}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-181}) \).
Endomorphism algebra over $\overline{\F}_{181}$
The base change of $A$ to $\F_{181^{2}}$ is the simple isogeny class 1.32761.ny and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $181$ and $\infty$.

Base change

This is a primitive isogeny class.

Twists

This isogeny class has no twists.