Properties

Label 2.61.ay_kf
Base Field $\F_{61}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{61}$
Dimension:  $2$
L-polynomial:  $( 1 - 13 x + 61 x^{2} )( 1 - 11 x + 61 x^{2} )$
Frobenius angles:  $\pm0.187058313935$, $\pm0.251304563322$
Angle rank:  $2$ (numerical)
Jacobians:  14

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 14 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2499 13682025 51717064896 191882861829225 713424990133573179 2654370081056816640000 9876830711753278313754819 36751688609722556947197158025 136753049740527961148134288933056 508858108562305110275038089556925625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 38 3676 227846 13858516 844693478 51520784038 3142742256398 191707285627876 11694145827742526 713342910180602476

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The isogeny class factors as 1.61.an $\times$ 1.61.al 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 $\F_{61}$.

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_av$2$(not in LMFDB)
2.61.c_av$2$(not in LMFDB)
2.61.y_kf$2$(not in LMFDB)
2.61.am_fd$3$(not in LMFDB)
2.61.d_abg$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.61.ac_av$2$(not in LMFDB)
2.61.c_av$2$(not in LMFDB)
2.61.y_kf$2$(not in LMFDB)
2.61.am_fd$3$(not in LMFDB)
2.61.d_abg$3$(not in LMFDB)
2.61.az_kq$6$(not in LMFDB)
2.61.ak_eh$6$(not in LMFDB)
2.61.ad_abg$6$(not in LMFDB)
2.61.k_eh$6$(not in LMFDB)
2.61.m_fd$6$(not in LMFDB)
2.61.z_kq$6$(not in LMFDB)