Properties

Label 2.5.ae_l
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:  $1 - 4 x + 11 x^{2} - 20 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.185749715683$, $\pm0.480916950984$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{12})\)
Galois group:  $C_2^2$
Jacobians:  2

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 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 13 793 16900 381433 10005853 251539600 6147184693 152176891113 3809093856100 95362793192953

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 32 134 612 3202 16094 78682 389572 1950254 9765152

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\).
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{3}}$ is 1.125.e 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
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.e_l$2$2.25.g_l
2.5.i_ba$3$2.125.i_kg
2.5.ac_ab$4$2.625.ao_aqn
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.e_l$2$2.25.g_l
2.5.i_ba$3$2.125.i_kg
2.5.ac_ab$4$2.625.ao_aqn
2.5.c_ab$4$2.625.ao_aqn
2.5.ai_ba$6$(not in LMFDB)
2.5.a_ag$6$(not in LMFDB)
2.5.ag_s$12$(not in LMFDB)
2.5.ae_o$12$(not in LMFDB)
2.5.ac_c$12$(not in LMFDB)
2.5.a_g$12$(not in LMFDB)
2.5.c_c$12$(not in LMFDB)
2.5.e_o$12$(not in LMFDB)
2.5.g_s$12$(not in LMFDB)
2.5.a_ai$24$(not in LMFDB)
2.5.a_i$24$(not in LMFDB)