Properties

Label 1.449.h
Base Field $\F_{449}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{449}$
Dimension:  $1$
L-polynomial:  $1 + 7 x + 449 x^{2}$
Frobenius angles:  $\pm0.552818961438$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-1747}) \)
Galois group:  $C_2$
Jacobians:  5

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 the Jacobians of 5 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 457 202451 90509764 40642645603 18248696780057 8193662122767104 3678954245384539193 1651850457738256968003 741680855534987499172036 333014704134435107391483251

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 457 202451 90509764 40642645603 18248696780057 8193662122767104 3678954245384539193 1651850457738256968003 741680855534987499172036 333014704134435107391483251

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{449}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1747}) \).
All geometric endomorphisms are defined over $\F_{449}$.

Base change

This is a primitive isogeny class.

Twists

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