Properties

Label 2.49.ay_jg
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:  $1 - 24 x + 240 x^{2} - 1176 x^{3} + 2401 x^{4}$
Frobenius angles:  $\pm0.0924052878182$, $\pm0.227088916785$
Angle rank:  $2$ (numerical)
Number field:  4.0.86272.3
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1442 5540164 13832888258 33248452141072 79799785101333122 191583148818026007364 459986723393504565729506 1104427614494690973889855488 2651730829144741159458181992866 6366805774994180929409558304290884

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 26 2306 117578 5767494 282501866 13841425730 678223348346 33232928771710 1628413587645914 79792266474134786

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.86272.3.
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.y_jg$2$(not in LMFDB)