Properties

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

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Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x )^{2}( 1 - 8 x + 49 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.306389419005$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1512 5612544 13838478696 33226260480000 79781820476078952 191575128888463574016 459984415854857021776872 1104427232468159779307520000 2651730809531246043410493784296 6366805760003910324030652455650304

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 28 2338 117628 5763646 282438268 13840846306 678219946012 33232917276286 1628413575601372 79792266286268578

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The isogeny class factors as 1.49.ao $\times$ 1.49.ai 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.ag_ao$2$(not in LMFDB)
2.49.g_ao$2$(not in LMFDB)
2.49.w_ic$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.ag_ao$2$(not in LMFDB)
2.49.g_ao$2$(not in LMFDB)
2.49.w_ic$2$(not in LMFDB)
2.49.ai_du$4$(not in LMFDB)
2.49.i_du$4$(not in LMFDB)