Properties

Label 2.97.a_cg
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:  $1 + 58 x^{2} + 9409 x^{4}$
Frobenius angles:  $\pm0.298321371491$, $\pm0.701678628509$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-7}, \sqrt{34})\)
Galois group:  $C_2^2$
Jacobians:  $740$
Cyclic group of points:    no
Non-cyclic primes:   $2, 3$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $9468$ $89643024$ $832970562876$ $7840170273309696$ $73742412706643630268$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $98$ $9526$ $912674$ $88560190$ $8587340258$ $832969120822$ $80798284478114$ $7837433470841854$ $760231058654565218$ $73742412723794434486$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 740 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 \(\Q(\sqrt{-7}, \sqrt{34})\).
Endomorphism algebra over $\overline{\F}_{97}$
The base change of $A$ to $\F_{97^{2}}$ is 1.9409.cg 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-238}) \)$)$

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_acg$4$(not in LMFDB)