Properties

Label 2.7.ad_e
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:  $( 1 - 5 x + 7 x^{2} )( 1 + 2 x + 7 x^{2} )$
  $1 - 3 x + 4 x^{2} - 21 x^{3} + 49 x^{4}$
Frobenius angles:  $\pm0.106147807505$, $\pm0.623375857214$
Angle rank:  $2$ (numerical)
Jacobians:  $2$
Isomorphism classes:  18

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $30$ $2340$ $100440$ $5709600$ $287002650$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $49$ $290$ $2377$ $17075$ $117466$ $824045$ $5773873$ $40361870$ $282483289$

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 $\F_{7}$.

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 1.7.c 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
2.7.ah_y$2$2.49.ab_am
2.7.d_e$2$2.49.ab_am
2.7.h_y$2$2.49.ab_am
2.7.d_q$3$2.343.acc_cao
2.7.g_w$3$2.343.acc_cao

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

TwistExtension degreeCommon base change
2.7.ah_y$2$2.49.ab_am
2.7.d_e$2$2.49.ab_am
2.7.h_y$2$2.49.ab_am
2.7.d_q$3$2.343.acc_cao
2.7.g_w$3$2.343.acc_cao
2.7.ag_w$6$(not in LMFDB)
2.7.ad_q$6$(not in LMFDB)
2.7.ac_g$6$(not in LMFDB)
2.7.ab_m$6$(not in LMFDB)
2.7.b_m$6$(not in LMFDB)
2.7.c_g$6$(not in LMFDB)