Properties

Label 2.61.aba_le
Base field $\F_{61}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{61}$
Dimension:  $2$
L-polynomial:  $( 1 - 14 x + 61 x^{2} )( 1 - 12 x + 61 x^{2} )$
  $1 - 26 x + 290 x^{2} - 1586 x^{3} + 3721 x^{4}$
Frobenius angles:  $\pm0.146275019398$, $\pm0.221142061624$
Angle rank:  $2$ (numerical)
Jacobians:  $10$
Isomorphism classes:  40

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2400$ $13497600$ $51585660000$ $191830914662400$ $713421947728860000$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $36$ $3626$ $227268$ $13854766$ $844689876$ $51521030138$ $3142745838996$ $191707316101726$ $11694145994559108$ $713342910530017226$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$
The isogeny class factors as 1.61.ao $\times$ 1.61.am and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.61.ac_abu$2$(not in LMFDB)
2.61.c_abu$2$(not in LMFDB)
2.61.ba_le$2$(not in LMFDB)
2.61.al_eg$3$(not in LMFDB)
2.61.b_abi$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.61.ac_abu$2$(not in LMFDB)
2.61.c_abu$2$(not in LMFDB)
2.61.ba_le$2$(not in LMFDB)
2.61.al_eg$3$(not in LMFDB)
2.61.b_abi$3$(not in LMFDB)
2.61.ay_kc$4$(not in LMFDB)
2.61.ae_as$4$(not in LMFDB)
2.61.e_as$4$(not in LMFDB)
2.61.y_kc$4$(not in LMFDB)
2.61.az_ks$6$(not in LMFDB)
2.61.an_fe$6$(not in LMFDB)
2.61.ab_abi$6$(not in LMFDB)
2.61.l_eg$6$(not in LMFDB)
2.61.n_fe$6$(not in LMFDB)
2.61.z_ks$6$(not in LMFDB)
2.61.ax_js$12$(not in LMFDB)
2.61.al_fc$12$(not in LMFDB)
2.61.aj_ei$12$(not in LMFDB)
2.61.ad_ai$12$(not in LMFDB)
2.61.d_ai$12$(not in LMFDB)
2.61.j_ei$12$(not in LMFDB)
2.61.l_fc$12$(not in LMFDB)
2.61.x_js$12$(not in LMFDB)