Properties

Label 2.5.af_p
Base Field $\F_{5}$
Dimension $2$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $1 - 5 x + 15 x^{2} - 25 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.200000000000$, $\pm0.400000000000$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\zeta_{5})\)
Galois group:  $C_4$
Jacobians:  2

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2, 1/2, 1/2]$

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 11 781 19151 406901 9759376 246109501 6152578751 152832422501 3808599606251 95245419909376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1 31 151 651 3126 15751 78751 391251 1950001 9753126

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{5})\).
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{10}}$ is 1.9765625.ajgk 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
All geometric endomorphisms are defined over $\F_{5^{10}}$.
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
2.5.a_ak$5$(not in LMFDB)
2.5.f_p$5$(not in LMFDB)
2.5.a_f$15$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.a_ak$5$(not in LMFDB)
2.5.f_p$5$(not in LMFDB)
2.5.a_f$15$(not in LMFDB)
2.5.a_k$20$(not in LMFDB)
2.5.a_f$30$(not in LMFDB)
2.5.a_a$40$(not in LMFDB)
2.5.a_af$60$(not in LMFDB)