Properties

Label 2.5.ab_ae
Base field $\F_{5}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple no
Primitive yes
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:  $1$
Isomorphism classes:  3

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $16$ $448$ $12544$ $410368$ $9456976$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{3}}$.

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 $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

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)