Properties

Label 3.5.ah_ba_acq
Base field $\F_{5}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{5}$
Dimension:  $3$
L-polynomial:  $1 - 7 x + 26 x^{2} - 68 x^{3} + 130 x^{4} - 175 x^{5} + 125 x^{6}$
Frobenius angles:  $\pm0.0605820805461$, $\pm0.287119770935$, $\pm0.511693182811$
Angle rank:  $3$ (numerical)
Number field:  6.0.2419571.1
Galois group:  $S_4\times C_2$
Jacobians:  $0$
Isomorphism classes:  4

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $32$ $17024$ $1936928$ $228053504$ $30324608512$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $29$ $125$ $585$ $3104$ $15605$ $77293$ $388625$ $1955159$ $9779244$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5}$.

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is 6.0.2419571.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
3.5.h_ba_cq$2$3.25.d_aq_acm