Properties

Label 2.61.ay_jt
Base Field $\F_{61}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{61}$
Dimension:  $2$
L-polynomial:  $1 - 24 x + 253 x^{2} - 1464 x^{3} + 3721 x^{4}$
Frobenius angles:  $\pm0.0139265954990$, $\pm0.319406737834$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{13})\)
Galois group:  $C_2^2$
Jacobians:  8

This isogeny class is simple but not 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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2487 13586481 51519935952 191672109482841 713280258809833287 2654303800498182146304 9876813687554472930864063 36751689786707188701387599529 136753052840547989392630633704912 508858109516080801856437157732296401

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 38 3652 226982 13843300 844522118 51519497542 3142736839406 191707291767364 11694146092834142 713342911517653252

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{61}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{13})\).
Endomorphism algebra over $\overline{\F}_{61}$
The base change of $A$ to $\F_{61^{6}}$ is 1.51520374361.ayyny 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$
All geometric endomorphisms are defined over $\F_{61^{6}}$.
Remainder of endomorphism lattice by field

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_cs$3$(not in LMFDB)
2.61.y_jt$3$(not in LMFDB)
2.61.a_cs$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.61.a_cs$3$(not in LMFDB)
2.61.y_jt$3$(not in LMFDB)
2.61.a_cs$6$(not in LMFDB)
2.61.a_acs$12$(not in LMFDB)