Properties

Label 2.49.aw_ih
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:  $( 1 - 13 x + 49 x^{2} )( 1 - 9 x + 49 x^{2} )$
Frobenius angles:  $\pm0.121037718324$, $\pm0.277748883973$
Angle rank:  $2$ (numerical)
Jacobians:  16

This isogeny class is not 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 16 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1517 5638689 13877588816 33257579784345 79798599518267477 191581729178259603456 459986427486813036231077 1104427761809799742219228905 2651730972034437890615161956176 6366805831001365816387225445552529

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 28 2348 117958 5769076 282497668 13841323166 678222912052 33232933204516 1628413675393702 79792267176047228

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The isogeny class factors as 1.49.an $\times$ 1.49.aj 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_at$2$(not in LMFDB)
2.49.e_at$2$(not in LMFDB)
2.49.w_ih$2$(not in LMFDB)
2.49.ah_dc$3$(not in LMFDB)
2.49.c_ab$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.ae_at$2$(not in LMFDB)
2.49.e_at$2$(not in LMFDB)
2.49.w_ih$2$(not in LMFDB)
2.49.ah_dc$3$(not in LMFDB)
2.49.c_ab$3$(not in LMFDB)
2.49.au_hp$6$(not in LMFDB)
2.49.al_em$6$(not in LMFDB)
2.49.ac_ab$6$(not in LMFDB)
2.49.h_dc$6$(not in LMFDB)
2.49.l_em$6$(not in LMFDB)
2.49.u_hp$6$(not in LMFDB)