Properties

Label 1.277.i
Base field $\F_{277}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{277}$
Dimension:  $1$
L-polynomial:  $1 + 8 x + 277 x^{2}$
Frobenius angles:  $\pm0.577257855233$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-29}) \)
Galois group:  $C_2$
Jacobians:  $18$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $286$ $77220$ $21247798$ $5887252800$ $1630795417966$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $286$ $77220$ $21247798$ $5887252800$ $1630795417966$ $451729672825860$ $125129117325604678$ $34660765697822035200$ $9601032097274730272446$ $2659485890898255690800100$

Jacobians and polarizations

This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{277}$.

Endomorphism algebra over $\F_{277}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-29}) \).

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
1.277.ai$2$(not in LMFDB)