Properties

Label 3.19.g_ag_afo
Base field $\F_{19}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes

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Invariants

Base field:  $\F_{19}$
Dimension:  $3$
L-polynomial:  $1 + 6 x - 6 x^{2} - 144 x^{3} - 114 x^{4} + 2166 x^{5} + 6859 x^{6}$
Frobenius angles:  $\pm0.138106922050$, $\pm0.731501495324$, $\pm0.880842547929$
Angle rank:  $3$ (numerical)
Number field:  6.0.379750680000.1
Galois group:  $S_4\times C_2$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $8768$ $41314816$ $317699431424$ $2226414615201792$ $15177128498780571968$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $26$ $314$ $6752$ $131090$ $2475446$ $47061548$ $893934914$ $16983834530$ $322687706528$ $6131074866074$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{19}$.

Endomorphism algebra over $\F_{19}$
The endomorphism algebra of this simple isogeny class is 6.0.379750680000.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
3.19.ag_ag_fo$2$(not in LMFDB)