Properties

Label 1.625.abq
Base Field $\F_{5^{4}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{5^{4}}$
Dimension:  $1$
L-polynomial:  $1 - 42 x + 625 x^{2}$
Frobenius angles:  $\pm0.182554890994$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-46}) \)
Galois group:  $C_2$

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 584 390112 244145288 152588407680 95367450443144 59604645241937632 37252902992462541128 23283064365424793832960 14551915228363538580904904 9094947017729119579330531552

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 584 390112 244145288 152588407680 95367450443144 59604645241937632 37252902992462541128 23283064365424793832960 14551915228363538580904904 9094947017729119579330531552

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5^{4}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-46}) \).
All geometric endomorphisms are defined over $\F_{5^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.625.bq$2$(not in LMFDB)