Properties

Label 2.49.ay_jh
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:  $( 1 - 13 x + 49 x^{2} )( 1 - 11 x + 49 x^{2} )$
Frobenius angles:  $\pm0.121037718324$, $\pm0.212295615010$
Angle rank:  $1$ (numerical)
Jacobians:  10

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 10 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1443 5545449 13841440704 33256196289225 79804666755947523 191585480762348015616 459987576359836495764963 1104427831886110649837126025 2651730845859650714729969939904 6366805754738231238590052255431529

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 26 2308 117650 5768836 282519146 13841594206 678224605994 33232935313156 1628413597910450 79792266220276228

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The isogeny class factors as 1.49.an $\times$ 1.49.al and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{7^{2}}$
The base change of $A$ to $\F_{7^{12}}$ is 1.13841287201.itby 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
All geometric endomorphisms are defined over $\F_{7^{12}}$.
Remainder of endomorphism lattice by field

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.ac_abt$2$(not in LMFDB)
2.49.c_abt$2$(not in LMFDB)
2.49.y_jh$2$(not in LMFDB)
2.49.ap_eu$3$(not in LMFDB)
2.49.aj_cy$3$(not in LMFDB)
2.49.a_act$3$(not in LMFDB)
2.49.a_ax$3$(not in LMFDB)
2.49.a_dq$3$(not in LMFDB)
2.49.j_cy$3$(not in LMFDB)
2.49.p_eu$3$(not in LMFDB)
2.49.y_jh$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.ac_abt$2$(not in LMFDB)
2.49.c_abt$2$(not in LMFDB)
2.49.y_jh$2$(not in LMFDB)
2.49.ap_eu$3$(not in LMFDB)
2.49.aj_cy$3$(not in LMFDB)
2.49.a_act$3$(not in LMFDB)
2.49.a_ax$3$(not in LMFDB)
2.49.a_dq$3$(not in LMFDB)
2.49.j_cy$3$(not in LMFDB)
2.49.p_eu$3$(not in LMFDB)
2.49.y_jh$3$(not in LMFDB)
2.49.aba_kh$6$(not in LMFDB)
2.49.aw_il$6$(not in LMFDB)
2.49.an_eq$6$(not in LMFDB)
2.49.al_cu$6$(not in LMFDB)
2.49.ae_dy$6$(not in LMFDB)
2.49.e_dy$6$(not in LMFDB)
2.49.l_cu$6$(not in LMFDB)
2.49.n_eq$6$(not in LMFDB)
2.49.w_il$6$(not in LMFDB)
2.49.ba_kh$6$(not in LMFDB)
2.49.a_adq$12$(not in LMFDB)
2.49.a_x$12$(not in LMFDB)
2.49.a_ct$12$(not in LMFDB)