Properties

Label 2.113.abq_zr
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

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Invariants

Base field:  $\F_{113}$
Dimension:  $2$
L-polynomial:  $( 1 - 21 x + 113 x^{2} )^{2}$
  $1 - 42 x + 667 x^{2} - 4746 x^{3} + 12769 x^{4}$
Frobenius angles:  $\pm0.0498602789898$, $\pm0.0498602789898$
Angle rank:  $1$ (numerical)
Jacobians:  $1$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $8649$ $157628025$ $2075777851536$ $26577696761555625$ $339449650075636504569$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $72$ $12340$ $1438614$ $163005988$ $18423967032$ $2081948347870$ $235260520023384$ $26584441725349828$ $3004041936872685942$ $339456738991898823700$

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 $\F_{113}$.

Endomorphism algebra over $\F_{113}$
The isogeny class factors as 1.113.av 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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.a_aih$2$(not in LMFDB)
2.113.bq_zr$2$(not in LMFDB)
2.113.v_mq$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.113.a_aih$2$(not in LMFDB)
2.113.bq_zr$2$(not in LMFDB)
2.113.v_mq$3$(not in LMFDB)
2.113.a_ih$4$(not in LMFDB)
2.113.av_mq$6$(not in LMFDB)