Properties

Label 2.7.af_o
Base field $\F_{7}$
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:  $\F_{7}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x + 7 x^{2} )( 1 + 7 x^{2} )$
  $1 - 5 x + 14 x^{2} - 35 x^{3} + 49 x^{4}$
Frobenius angles:  $\pm0.106147807505$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  $2$
Isomorphism classes:  10

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $24$ $2496$ $111456$ $5481216$ $282929064$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $53$ $324$ $2281$ $16833$ $118622$ $824799$ $5764273$ $40366188$ $282541853$

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 $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 1.7.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 $\overline{\F}_{7}$
The base change of $A$ to $\F_{7^{2}}$ is 1.49.al $\times$ 1.49.o. 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
2.7.f_o$2$2.49.d_ace
2.7.b_o$3$2.343.au_bak
2.7.e_o$3$2.343.au_bak

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

TwistExtension degreeCommon base change
2.7.f_o$2$2.49.d_ace
2.7.b_o$3$2.343.au_bak
2.7.e_o$3$2.343.au_bak
2.7.ae_o$6$(not in LMFDB)
2.7.ab_o$6$(not in LMFDB)