Properties

Label 2.7.ad_c
Base field $\F_{7}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
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:  $1 - 3 x + 2 x^{2} - 21 x^{3} + 49 x^{4}$
Frobenius angles:  $\pm0.0252087988121$, $\pm0.641457867855$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-19})\)
Galois group:  $C_2^2$
Jacobians:  $1$
Isomorphism classes:  2

This isogeny class is simple but not geometrically 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $28$ $2128$ $94864$ $5592384$ $281904868$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $45$ $272$ $2329$ $16775$ $116430$ $821945$ $5765329$ $40334384$ $282442725$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is 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 endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-19})\).
Endomorphism algebra over $\overline{\F}_{7}$
The base change of $A$ to $\F_{7^{3}}$ is 1.343.abk 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

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.d_c$2$2.49.af_ay
2.7.g_x$3$2.343.acu_cyg
2.7.ag_x$6$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.7.d_c$2$2.49.af_ay
2.7.g_x$3$2.343.acu_cyg
2.7.ag_x$6$(not in LMFDB)
2.7.a_f$6$(not in LMFDB)
2.7.a_af$12$(not in LMFDB)