Properties

Label 2.49.au_hg
Base field $\F_{7^{2}}$
Dimension $2$
$p$-rank $2$
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:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 20 x + 188 x^{2} - 980 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.110672799774$, $\pm0.337577521471$
Angle rank:  $2$ (numerical)
Number field:  4.0.5433600.1
Galois group:  $D_{4}$
Jacobians:  $16$
Isomorphism classes:  32

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:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1590$ $5708100$ $13881286710$ $33240526707600$ $79788566331849750$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $30$ $2378$ $117990$ $5766118$ $282462150$ $13841168618$ $678223637310$ $33232948154878$ $1628413758738270$ $79792267046027498$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.5433600.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.49.u_hg$2$(not in LMFDB)