Properties

Label 2.61.abc_mf
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:  $( 1 - 15 x + 61 x^{2} )( 1 - 13 x + 61 x^{2} )$
Frobenius angles:  $\pm0.0900194921159$, $\pm0.187058313935$
Angle rank:  $2$ (numerical)
Jacobians:  7

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 7 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2303 13299825 51419025728 191735929625625 713383328457705983 2654373596011548364800 9876843243372879484117463 36751697301462218284028675625 136753053554405836488875124615488 508858109659159983994911334629020625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 34 3572 226534 13847908 844644154 51520852262 3142746243874 191707330966468 11694146153878174 713342911718228852

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The isogeny class factors as 1.61.ap $\times$ 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:
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_acv$2$(not in LMFDB)
2.61.c_acv$2$(not in LMFDB)
2.61.bc_mf$2$(not in LMFDB)
2.61.aq_fh$3$(not in LMFDB)
2.61.ab_adk$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.61.ac_acv$2$(not in LMFDB)
2.61.c_acv$2$(not in LMFDB)
2.61.bc_mf$2$(not in LMFDB)
2.61.aq_fh$3$(not in LMFDB)
2.61.ab_adk$3$(not in LMFDB)
2.61.abd_mu$6$(not in LMFDB)
2.61.ao_ed$6$(not in LMFDB)
2.61.b_adk$6$(not in LMFDB)
2.61.o_ed$6$(not in LMFDB)
2.61.q_fh$6$(not in LMFDB)
2.61.bd_mu$6$(not in LMFDB)