Properties

Label 2.121.abk_vl
Base field $\F_{11^{2}}$
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:  $\F_{11^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 21 x + 121 x^{2} )( 1 - 15 x + 121 x^{2} )$
  $1 - 36 x + 557 x^{2} - 4356 x^{3} + 14641 x^{4}$
Frobenius angles:  $\pm0.0963413489042$, $\pm0.261189521777$
Angle rank:  $2$ (numerical)
Jacobians:  $44$
Isomorphism classes:  80

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $10807$ $211719937$ $3139193843968$ $45953732930873625$ $672754298622262473727$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $86$ $14460$ $1771994$ $214377556$ $25937590526$ $3138428495022$ $379749820441406$ $45949729773931876$ $5559917315671205354$ $672749995001954839980$

Jacobians and polarizations

This isogeny class contains the Jacobians of 44 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{2}}$.

Endomorphism algebra over $\F_{11^{2}}$
The isogeny class factors as 1.121.av $\times$ 1.121.ap 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 is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.121.ag_acv$2$(not in LMFDB)
2.121.g_acv$2$(not in LMFDB)
2.121.bk_vl$2$(not in LMFDB)