Properties

Label 2.97.abi_so
Base field $\F_{97}$
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:  $\F_{97}$
Dimension:  $2$
L-polynomial:  $( 1 - 18 x + 97 x^{2} )( 1 - 16 x + 97 x^{2} )$
  $1 - 34 x + 482 x^{2} - 3298 x^{3} + 9409 x^{4}$
Frobenius angles:  $\pm0.133124938748$, $\pm0.198227810371$
Angle rank:  $2$ (numerical)
Jacobians:  $16$
Isomorphism classes:  60

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6560$ $86749440$ $832942466720$ $7838929236787200$ $73744794903449712800$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $64$ $9218$ $912640$ $88546174$ $8587617664$ $832974989186$ $80798308262848$ $7837433720594686$ $760231058713689280$ $73742412681029289218$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97}$.

Endomorphism algebra over $\F_{97}$
The isogeny class factors as 1.97.as $\times$ 1.97.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
2.97.ac_adq$2$(not in LMFDB)
2.97.c_adq$2$(not in LMFDB)
2.97.bi_so$2$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.97.ac_adq$2$(not in LMFDB)
2.97.c_adq$2$(not in LMFDB)
2.97.bi_so$2$(not in LMFDB)
2.97.ay_mk$4$(not in LMFDB)
2.97.ai_co$4$(not in LMFDB)
2.97.i_co$4$(not in LMFDB)
2.97.y_mk$4$(not in LMFDB)