Properties

Label 2.5.ad_e
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:  $1 - 3 x + 4 x^{2} - 15 x^{3} + 25 x^{4}$
Frobenius angles:  $\pm0.0673911931187$, $\pm0.599275473548$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-11})\)
Galois group:  $C_2^2$
Jacobians:  3

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 3 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 12 576 11664 361728 9943932 241864704 6064084668 153038435328 3818287953936 95338124237376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 25 90 577 3183 15478 77619 391777 1954962 9762625

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\).
Endomorphism algebra over $\overline{\F}_{5}$
The base change of $A$ to $\F_{5^{3}}$ is 1.125.as 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
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.d_e$2$2.25.ab_ay
2.5.g_t$3$2.125.abk_wc
2.5.ag_t$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.5.d_e$2$2.25.ab_ay
2.5.g_t$3$2.125.abk_wc
2.5.ag_t$6$(not in LMFDB)
2.5.a_b$6$(not in LMFDB)
2.5.d_e$6$(not in LMFDB)
2.5.a_ab$12$(not in LMFDB)