Properties

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

Learn more about

Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x )^{2}( 1 - 6 x + 49 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.359017035971$
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
$A(\F_{q^r})$ 1584 5677056 13838712624 33210777600000 79775178013683504 191575092071265509376 459985465482543519807024 1104427643743877470617600000 2651730812702512974796073358384 6366805703230630314611468677718016

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 30 2366 117630 5760958 282414750 13840843646 678221493630 33232929651838 1628413577548830 79792265574755006

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The isogeny class factors as 1.49.ao $\times$ 1.49.ag 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.ai_o$2$(not in LMFDB)
2.49.i_o$2$(not in LMFDB)
2.49.u_ha$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.ai_o$2$(not in LMFDB)
2.49.i_o$2$(not in LMFDB)
2.49.u_ha$2$(not in LMFDB)
2.49.ag_du$4$(not in LMFDB)
2.49.g_du$4$(not in LMFDB)