Properties

Label 2.49.ay_je
Base Field $\F_{7^{2}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x )^{2}( 1 - 10 x + 49 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.246751714429$
Angle rank:  $1$ (numerical)
Jacobians:  6

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1440 5529600 13815787680 33232896000000 79789818658327200 191578173864518246400 459984687945085052629920 1104426909195338612736000000 2651730613372185739653797292960 6366805711238142926910784650240000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 26 2302 117434 5764798 282466586 13841066302 678220347194 33232907548798 1628413455141146 79792265675109502

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The isogeny class factors as 1.49.ao $\times$ 1.49.ak and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{7^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.ae_abq$2$(not in LMFDB)
2.49.e_abq$2$(not in LMFDB)
2.49.y_je$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.ae_abq$2$(not in LMFDB)
2.49.e_abq$2$(not in LMFDB)
2.49.y_je$2$(not in LMFDB)
2.49.ak_du$4$(not in LMFDB)
2.49.k_du$4$(not in LMFDB)