Properties

Label 2.61.az_km
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:  $( 1 - 15 x + 61 x^{2} )( 1 - 10 x + 61 x^{2} )$
Frobenius angles:  $\pm0.0900194921159$, $\pm0.278857938376$
Angle rank:  $2$ (numerical)
Jacobians:  15

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 15 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2444 13549536 51565701824 191759808240000 713351253568586204 2654339810149110859776 9876825946723658494814684 36751693269957700872277440000 136753055010413209795368992732864 508858111714715247375944979681371616

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 37 3641 227182 13849633 844606177 51520196486 3142740740197 191707309936993 11694146278385542 713342914599809681

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The isogeny class factors as 1.61.ap $\times$ 1.61.ak 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.af_abc$2$(not in LMFDB)
2.61.f_abc$2$(not in LMFDB)
2.61.z_km$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.61.af_abc$2$(not in LMFDB)
2.61.f_abc$2$(not in LMFDB)
2.61.z_km$2$(not in LMFDB)
2.61.abb_lq$4$(not in LMFDB)
2.61.ad_acg$4$(not in LMFDB)
2.61.d_acg$4$(not in LMFDB)
2.61.bb_lq$4$(not in LMFDB)