Properties

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

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2209 13097161 51235227904 191619651155625 713327584011549529 2654354854115086667776 9876841392793072459507729 36751700628427917473879855625 136753057025814888000782067437824 508858111912130669426735905295937241

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 32 3516 225722 13839508 844578152 51520488486 3142745655032 191707348320868 11694146450728322 713342914876556556

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The isogeny class factors as 1.61.ap 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$
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.a_adz$2$(not in LMFDB)
2.61.be_nj$2$(not in LMFDB)
2.61.p_gi$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.61.a_adz$2$(not in LMFDB)
2.61.be_nj$2$(not in LMFDB)
2.61.p_gi$3$(not in LMFDB)
2.61.a_dz$4$(not in LMFDB)
2.61.ap_gi$6$(not in LMFDB)