Properties

Label 2.113.abk_va
Base field $\F_{113}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{113}$
Dimension:  $2$
L-polynomial:  $( 1 - 20 x + 113 x^{2} )( 1 - 16 x + 113 x^{2} )$
  $1 - 36 x + 546 x^{2} - 4068 x^{3} + 12769 x^{4}$
Frobenius angles:  $\pm0.110150159186$, $\pm0.228810695365$
Angle rank:  $2$ (numerical)
Jacobians:  $36$
Isomorphism classes:  92

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $9212$ $160473040$ $2082108851228$ $26587686780928000$ $339462031324100340572$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $78$ $12566$ $1443006$ $163067262$ $18424639038$ $2081954272214$ $235260561489486$ $26584441951774078$ $3004041938042961198$ $339456739004455819286$

Jacobians and polarizations

This isogeny class contains the Jacobians of 36 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{113}$.

Endomorphism algebra over $\F_{113}$
The isogeny class factors as 1.113.au $\times$ 1.113.aq 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
2.113.ae_adq$2$(not in LMFDB)
2.113.e_adq$2$(not in LMFDB)
2.113.bk_va$2$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.113.ae_adq$2$(not in LMFDB)
2.113.e_adq$2$(not in LMFDB)
2.113.bk_va$2$(not in LMFDB)
2.113.abi_tm$4$(not in LMFDB)
2.113.ag_acc$4$(not in LMFDB)
2.113.g_acc$4$(not in LMFDB)
2.113.bi_tm$4$(not in LMFDB)