Properties

Label 2.61.abb_ls
Base Field $\F_{61}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{61}$
Dimension:  $2$
L-polynomial:  $( 1 - 14 x + 61 x^{2} )( 1 - 13 x + 61 x^{2} )$
Frobenius angles:  $\pm0.146275019398$, $\pm0.187058313935$
Angle rank:  $1$ (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
$A(\F_{q^r})$ 2352 13406400 51520795200 191807027193600 713423619261382512 2654392338040343040000 9876849924977249890820592 36751698483286618954697702400 136753052840548015382510835124800 508858108653623078737901572967160000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 35 3601 226982 13853041 844691855 51521216038 3142748369915 191707337131201 11694146092834142 713342910308616601

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The isogeny class factors as 1.61.ao $\times$ 1.61.an 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 $\overline{\F}_{61}$
The base change of $A$ to $\F_{61^{6}}$ is 1.51520374361.xyoc 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
All geometric endomorphisms are defined over $\F_{61^{6}}$.
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
2.61.ab_aci$2$(not in LMFDB)
2.61.b_aci$2$(not in LMFDB)
2.61.bb_ls$2$(not in LMFDB)
2.61.ap_fg$3$(not in LMFDB)
2.61.am_ef$3$(not in LMFDB)
2.61.a_acw$3$(not in LMFDB)
2.61.a_abv$3$(not in LMFDB)
2.61.a_er$3$(not in LMFDB)
2.61.m_ef$3$(not in LMFDB)
2.61.p_fg$3$(not in LMFDB)
2.61.bb_ls$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.61.ab_aci$2$(not in LMFDB)
2.61.b_aci$2$(not in LMFDB)
2.61.bb_ls$2$(not in LMFDB)
2.61.ap_fg$3$(not in LMFDB)
2.61.am_ef$3$(not in LMFDB)
2.61.a_acw$3$(not in LMFDB)
2.61.a_abv$3$(not in LMFDB)
2.61.a_er$3$(not in LMFDB)
2.61.m_ef$3$(not in LMFDB)
2.61.p_fg$3$(not in LMFDB)
2.61.bb_ls$3$(not in LMFDB)
2.61.abc_mg$6$(not in LMFDB)
2.61.aba_lf$6$(not in LMFDB)
2.61.ao_ff$6$(not in LMFDB)
2.61.an_ee$6$(not in LMFDB)
2.61.ac_et$6$(not in LMFDB)
2.61.c_et$6$(not in LMFDB)
2.61.n_ee$6$(not in LMFDB)
2.61.o_ff$6$(not in LMFDB)
2.61.ba_lf$6$(not in LMFDB)
2.61.bc_mg$6$(not in LMFDB)
2.61.a_aer$12$(not in LMFDB)
2.61.a_bv$12$(not in LMFDB)
2.61.a_cw$12$(not in LMFDB)