Properties

Label 2.7.af_s
Base field $\F_{7}$
Dimension $2$
$p$-rank $2$
Ordinary yes
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:  $( 1 - 4 x + 7 x^{2} )( 1 - x + 7 x^{2} )$
  $1 - 5 x + 18 x^{2} - 35 x^{3} + 49 x^{4}$
Frobenius angles:  $\pm0.227185525829$, $\pm0.439481140838$
Angle rank:  $1$ (numerical)
Jacobians:  $3$
Isomorphism classes:  14

This isogeny class is not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $28$ $3024$ $132496$ $5818176$ $282879268$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $3$ $61$ $384$ $2425$ $16833$ $118222$ $824799$ $5760529$ $40328448$ $282442261$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{3}}$.

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.ae $\times$ 1.7.ab 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 $\F_{7^{3}}$ is 1.343.u 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.7.ad_k$2$2.49.l_cu
2.7.d_k$2$2.49.l_cu
2.7.f_s$2$2.49.l_cu
2.7.ai_be$3$2.343.bo_bpu
2.7.ac_p$3$2.343.bo_bpu
2.7.b_ag$3$2.343.bo_bpu
2.7.e_j$3$2.343.bo_bpu
2.7.k_bn$3$2.343.bo_bpu

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

TwistExtension degreeCommon base change
2.7.ad_k$2$2.49.l_cu
2.7.d_k$2$2.49.l_cu
2.7.f_s$2$2.49.l_cu
2.7.ai_be$3$2.343.bo_bpu
2.7.ac_p$3$2.343.bo_bpu
2.7.b_ag$3$2.343.bo_bpu
2.7.e_j$3$2.343.bo_bpu
2.7.k_bn$3$2.343.bo_bpu
2.7.ak_bn$6$(not in LMFDB)
2.7.aj_bi$6$(not in LMFDB)
2.7.ag_t$6$(not in LMFDB)
2.7.ae_j$6$(not in LMFDB)
2.7.ab_ag$6$(not in LMFDB)
2.7.a_al$6$(not in LMFDB)
2.7.a_ac$6$(not in LMFDB)
2.7.a_n$6$(not in LMFDB)
2.7.c_p$6$(not in LMFDB)
2.7.g_t$6$(not in LMFDB)
2.7.i_be$6$(not in LMFDB)
2.7.j_bi$6$(not in LMFDB)
2.7.a_an$12$(not in LMFDB)
2.7.a_c$12$(not in LMFDB)
2.7.a_l$12$(not in LMFDB)