Properties

Label 2.25.ar_eq
Base field $\F_{5^{2}}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{5^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x )^{2}( 1 - 7 x + 25 x^{2} )$
  $1 - 17 x + 120 x^{2} - 425 x^{3} + 625 x^{4}$
Frobenius angles:  $0$, $0$, $\pm0.253183311107$
Angle rank:  $1$ (numerical)
Jacobians:  $0$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $304$ $361152$ $243063808$ $152586720000$ $95347337072944$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $9$ $577$ $15558$ $390625$ $9763569$ $244107502$ $6103241433$ $152586330625$ $3814690856694$ $95367414059377$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5^{2}}$
The isogeny class factors as 1.25.ak $\times$ 1.25.ah and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.25.ad_au$2$2.625.abx_bue
2.25.d_au$2$2.625.abx_bue
2.25.r_eq$2$2.625.abx_bue
2.25.ac_p$3$(not in LMFDB)

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.25.ad_au$2$2.625.abx_bue
2.25.d_au$2$2.625.abx_bue
2.25.r_eq$2$2.625.abx_bue
2.25.ac_p$3$(not in LMFDB)
2.25.am_dh$6$(not in LMFDB)
2.25.c_p$6$(not in LMFDB)
2.25.m_dh$6$(not in LMFDB)