Properties

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

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Invariants

Base field:  $\F_{7^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 11 x + 49 x^{2} )( 1 - 10 x + 49 x^{2} )$
Frobenius angles:  $\pm0.212295615010$, $\pm0.246751714429$
Angle rank:  $2$ (numerical)
Jacobians:  0

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 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
$A(\F_{q^r})$ 1560 5709600 13930600320 33285255120000 79808634019147800 191583555362580326400 459985853908391853109080 1104427068735782409433920000 2651730618514761281441921927040 6366805714559645619652482262740000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 29 2377 118406 5773873 282533189 13841455102 678222066341 33232912349473 1628413458299174 79792265716736377

Decomposition and endomorphism algebra

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