Properties

Label 2.61.abd_mt
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:  $1 - 29 x + 331 x^{2} - 1769 x^{3} + 3721 x^{4}$
Frobenius angles:  $\pm0.00565540645541$, $\pm0.172515385279$
Angle rank:  $2$ (numerical)
Number field:  \(\Q(\zeta_{5})\)
Galois group:  $C_4$
Jacobians:  1

This isogeny class is simple and geometrically 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
$A(\F_{q^r})$ 2255 13194005 51316721555 191662462946405 713338599476250000 2654348964974812242005 9876830332246605354975755 36751690553827489342421332805 136753049927269747265285817799655 508858107650663515605158797620000000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 33 3543 226083 13842603 844591198 51520374183 3142742135643 191707295768883 11694145843711353 713342908902617398

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{5})\).
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.bd_mt$2$(not in LMFDB)
2.61.b_abd$5$(not in LMFDB)
2.61.b_dn$5$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.61.bd_mt$2$(not in LMFDB)
2.61.b_abd$5$(not in LMFDB)
2.61.b_dn$5$(not in LMFDB)
2.61.l_bz$5$(not in LMFDB)
2.61.q_gk$5$(not in LMFDB)
2.61.aq_gk$10$(not in LMFDB)
2.61.al_bz$10$(not in LMFDB)
2.61.ab_abd$10$(not in LMFDB)
2.61.ab_dn$10$(not in LMFDB)