Properties

Label 2.61.az_kq
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:  $( 1 - 14 x + 61 x^{2} )( 1 - 11 x + 61 x^{2} )$
Frobenius angles:  $\pm0.146275019398$, $\pm0.251304563322$
Angle rank:  $2$ (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
$A(\F_{q^r})$ 2448 13581504 51634195200 191837603153664 713409532440053328 2654370081056816640000 9876835542770460249493968 36751693118511697468584459264 136753052498079096527966564524800 508858109809738883308964469679291584

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 37 3649 227482 13855249 844675177 51520784038 3142743793597 191707309147009 11694146063548642 713342911929317929

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The isogeny class factors as 1.61.ao $\times$ 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.ad_abg$2$(not in LMFDB)
2.61.d_abg$2$(not in LMFDB)
2.61.z_kq$2$(not in LMFDB)
2.61.ak_eh$3$(not in LMFDB)
2.61.c_av$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.61.ad_abg$2$(not in LMFDB)
2.61.d_abg$2$(not in LMFDB)
2.61.z_kq$2$(not in LMFDB)
2.61.ak_eh$3$(not in LMFDB)
2.61.c_av$3$(not in LMFDB)
2.61.ay_kf$6$(not in LMFDB)
2.61.am_fd$6$(not in LMFDB)
2.61.ac_av$6$(not in LMFDB)
2.61.k_eh$6$(not in LMFDB)
2.61.m_fd$6$(not in LMFDB)
2.61.y_kf$6$(not in LMFDB)