Properties

Label 2.49.aw_il
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:  $( 1 - 11 x + 49 x^{2} )^{2}$
Frobenius angles:  $\pm0.212295615010$, $\pm0.212295615010$
Angle rank:  $1$ (numerical)
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1521 5659641 13908900096 33282226355625 79810905102881121 191585480762348015616 459986634292967305197921 1104427227000104792784455625 2651730593295364763909071677696 6366805677394672646699363854447641

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 28 2356 118222 5773348 282541228 13841594206 678223216972 33232917111748 1628413442812078 79792265250964756

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7^{2}}$
The isogeny class factors as 1.49.al 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
All geometric endomorphisms are defined over $\F_{7^{2}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{2}}$.

SubfieldPrimitive Model
$\F_{7}$2.7.ak_bn
$\F_{7}$2.7.a_al
$\F_{7}$2.7.k_bn

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.a_ax$2$(not in LMFDB)
2.49.w_il$2$(not in LMFDB)
2.49.an_eq$3$(not in LMFDB)
2.49.ae_dy$3$(not in LMFDB)
2.49.c_abt$3$(not in LMFDB)
2.49.l_cu$3$(not in LMFDB)
2.49.ba_kh$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.49.a_ax$2$(not in LMFDB)
2.49.w_il$2$(not in LMFDB)
2.49.an_eq$3$(not in LMFDB)
2.49.ae_dy$3$(not in LMFDB)
2.49.c_abt$3$(not in LMFDB)
2.49.l_cu$3$(not in LMFDB)
2.49.ba_kh$3$(not in LMFDB)
2.49.a_x$4$(not in LMFDB)
2.49.aba_kh$6$(not in LMFDB)
2.49.ay_jh$6$(not in LMFDB)
2.49.ap_eu$6$(not in LMFDB)
2.49.al_cu$6$(not in LMFDB)
2.49.aj_cy$6$(not in LMFDB)
2.49.ac_abt$6$(not in LMFDB)
2.49.a_act$6$(not in LMFDB)
2.49.a_dq$6$(not in LMFDB)
2.49.e_dy$6$(not in LMFDB)
2.49.j_cy$6$(not in LMFDB)
2.49.n_eq$6$(not in LMFDB)
2.49.p_eu$6$(not in LMFDB)
2.49.y_jh$6$(not in LMFDB)
2.49.a_adq$12$(not in LMFDB)
2.49.a_ct$12$(not in LMFDB)