Properties

Label 2.5.ab_ae
Base Field $\F_{5}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{5}$
Dimension:  $2$
L-polynomial:  $1 - x - 4 x^{2} - 5 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.0948835201023$, $\pm0.761550186769$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-19})\)
Galois group:  $C_2^2$
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 16 448 12544 410368 9456976 245862400 6147422416 152474692608 3824496519424 95406021473728

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 17 98 657 3025 15734 78685 390337 1958138 9769577

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-19})\).
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{3}}$ is 1.125.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$
All geometric endomorphisms are defined over $\F_{5^{3}}$.

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.b_ae$2$2.25.aj_ce
2.5.c_l$3$2.125.abc_re
2.5.ac_l$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.b_ae$2$2.25.aj_ce
2.5.c_l$3$2.125.abc_re
2.5.ac_l$6$(not in LMFDB)
2.5.a_j$6$(not in LMFDB)
2.5.a_aj$12$(not in LMFDB)