Properties

Label 2.13.ah_ba
Base field $\F_{13}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{13}$
Dimension:  $2$
L-polynomial:  $( 1 - 7 x + 13 x^{2} )( 1 + 13 x^{2} )$
  $1 - 7 x + 26 x^{2} - 91 x^{3} + 169 x^{4}$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  $2$
Isomorphism classes:  18

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $98$ $28812$ $4677344$ $800743104$ $137700691898$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $7$ $173$ $2128$ $28033$ $370867$ $4830698$ $62750527$ $815694241$ $10604617744$ $137859794693$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{2}}$.

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 1.13.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{13}$
The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 1.169.ba. 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
2.13.h_ba$2$2.169.d_aka
2.13.c_ba$3$(not in LMFDB)
2.13.f_ba$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.13.h_ba$2$2.169.d_aka
2.13.c_ba$3$(not in LMFDB)
2.13.f_ba$3$(not in LMFDB)
2.13.af_ba$6$(not in LMFDB)
2.13.ac_ba$6$(not in LMFDB)