Properties

Label 2.49.av_hw
Base Field $\F_{7^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $1 - 21 x + 204 x^{2} - 1029 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.145558870031$, $\pm0.294073133631$
Angle rank:  $2$ (numerical)
Number field:  4.0.344488.2
Galois group:  $D_{4}$
Jacobians:  12

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

This isogeny class contains the Jacobians of 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1556 5688736 13900731392 33263563973248 79799379375799316 191581828815879307264 459986553392529725337428 1104427836212166485104200192 2651730978154050392541855822848 6366805811477623273783253635485856

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 29 2369 118154 5770113 282500429 13841330366 678223097693 33232935443329 1628413679151722 79792266931365089

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.344488.2.
All geometric endomorphisms are defined over $\F_{7^{2}}$.

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.v_hw$2$(not in LMFDB)