Properties

Label 2.97.abm_vj
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:  $( 1 - 19 x + 97 x^{2} )^{2}$
  $1 - 38 x + 555 x^{2} - 3686 x^{3} + 9409 x^{4}$
Frobenius angles:  $\pm0.0849741350078$, $\pm0.0849741350078$
Angle rank:  $1$ (numerical)
Jacobians:  $2$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6241$ $85433049$ $830547886336$ $7835827755484521$ $73741668375066892561$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $60$ $9076$ $910014$ $88511140$ $8587253580$ $832972117822$ $80798295030828$ $7837433783928004$ $760231061232421758$ $73742412720085649236$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 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.at 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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.a_agl$2$(not in LMFDB)
2.97.bm_vj$2$(not in LMFDB)
2.97.ao_dv$3$(not in LMFDB)
2.97.af_acu$3$(not in LMFDB)
2.97.k_il$3$(not in LMFDB)
2.97.t_ke$3$(not in LMFDB)
2.97.bc_pa$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.97.a_agl$2$(not in LMFDB)
2.97.bm_vj$2$(not in LMFDB)
2.97.ao_dv$3$(not in LMFDB)
2.97.af_acu$3$(not in LMFDB)
2.97.k_il$3$(not in LMFDB)
2.97.t_ke$3$(not in LMFDB)
2.97.bc_pa$3$(not in LMFDB)
2.97.a_gl$4$(not in LMFDB)
2.97.abh_rs$6$(not in LMFDB)
2.97.abc_pa$6$(not in LMFDB)
2.97.ay_ld$6$(not in LMFDB)
2.97.at_ke$6$(not in LMFDB)
2.97.ak_il$6$(not in LMFDB)
2.97.aj_eu$6$(not in LMFDB)
2.97.a_ac$6$(not in LMFDB)
2.97.a_gn$6$(not in LMFDB)
2.97.f_acu$6$(not in LMFDB)
2.97.j_eu$6$(not in LMFDB)
2.97.o_dv$6$(not in LMFDB)
2.97.y_ld$6$(not in LMFDB)
2.97.bh_rs$6$(not in LMFDB)
2.97.a_agn$12$(not in LMFDB)
2.97.a_c$12$(not in LMFDB)