Properties

Label 2.49.ax_iu
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:  $( 1 - 13 x + 49 x^{2} )( 1 - 10 x + 49 x^{2} )$
Frobenius angles:  $\pm0.121037718324$, $\pm0.246751714429$
Angle rank:  $2$ (numerical)
Jacobians:  9

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 9 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1480 5594400 13863035680 33259222684800 79802395849731400 191583555362580326400 459986795973662791903240 1104427673621701586376307200 2651730871079049634286196796960 6366805791903204663020946231060000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 27 2329 117834 5769361 282511107 13841455102 678223455363 33232930550881 1628413613397546 79792266686047849

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The isogeny class factors as 1.49.an $\times$ 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.ad_abg$2$(not in LMFDB)
2.49.d_abg$2$(not in LMFDB)
2.49.x_iu$2$(not in LMFDB)
2.49.ai_da$3$(not in LMFDB)
2.49.b_am$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.ad_abg$2$(not in LMFDB)
2.49.d_abg$2$(not in LMFDB)
2.49.x_iu$2$(not in LMFDB)
2.49.ai_da$3$(not in LMFDB)
2.49.b_am$3$(not in LMFDB)
2.49.av_ia$6$(not in LMFDB)
2.49.am_eo$6$(not in LMFDB)
2.49.ab_am$6$(not in LMFDB)
2.49.i_da$6$(not in LMFDB)
2.49.m_eo$6$(not in LMFDB)
2.49.v_ia$6$(not in LMFDB)