Properties

Label 2.97.a_cs
Base field $\F_{97}$
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:  $\F_{97}$
Dimension:  $2$
L-polynomial:  $1 + 70 x^{2} + 9409 x^{4}$
Frobenius angles:  $\pm0.308752374736$, $\pm0.691247625264$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{31}, \sqrt{-66})\)
Galois group:  $C_2^2$
Jacobians:  $848$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $9480$ $89870400$ $832970372040$ $7839898266240000$ $73742412706022339400$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $98$ $9550$ $912674$ $88557118$ $8587340258$ $832968739150$ $80798284478114$ $7837433561072638$ $760231058654565218$ $73742412722551852750$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 848 curves (of which all are hyperelliptic):

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97^{2}}$.

Endomorphism algebra over $\F_{97}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{31}, \sqrt{-66})\).
Endomorphism algebra over $\overline{\F}_{97}$
The base change of $A$ to $\F_{97^{2}}$ is 1.9409.cs 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2046}) \)$)$

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.97.a_acs$4$(not in LMFDB)