Properties

Label 2.79.a_fm
Base field $\F_{79}$
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:  $\F_{79}$
Dimension:  $2$
L-polynomial:  $( 1 - 4 x + 79 x^{2} )( 1 + 4 x + 79 x^{2} )$
  $1 + 142 x^{2} + 6241 x^{4}$
Frobenius angles:  $\pm0.427756044762$, $\pm0.572243955238$
Angle rank:  $1$ (numerical)
Jacobians:  $461$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6384$ $40755456$ $243087660144$ $1516510517760000$ $9468276078667841904$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $80$ $6526$ $493040$ $38934718$ $3077056400$ $243087864766$ $19203908986160$ $1517108847680638$ $119851595982618320$ $9468276074708836606$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 461 curves (of which all are hyperelliptic):

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{79^{2}}$.

Endomorphism algebra over $\F_{79}$
The isogeny class factors as 1.79.ae $\times$ 1.79.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{79}$
The base change of $A$ to $\F_{79^{2}}$ is 1.6241.fm 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
2.79.ai_gs$2$(not in LMFDB)
2.79.i_gs$2$(not in LMFDB)
2.79.abe_op$3$(not in LMFDB)
2.79.av_is$3$(not in LMFDB)
2.79.aj_ec$3$(not in LMFDB)
2.79.a_afb$3$(not in LMFDB)
2.79.a_al$3$(not in LMFDB)
2.79.j_ec$3$(not in LMFDB)
2.79.v_is$3$(not in LMFDB)
2.79.be_op$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.79.ai_gs$2$(not in LMFDB)
2.79.i_gs$2$(not in LMFDB)
2.79.abe_op$3$(not in LMFDB)
2.79.av_is$3$(not in LMFDB)
2.79.aj_ec$3$(not in LMFDB)
2.79.a_afb$3$(not in LMFDB)
2.79.a_al$3$(not in LMFDB)
2.79.j_ec$3$(not in LMFDB)
2.79.v_is$3$(not in LMFDB)
2.79.be_op$3$(not in LMFDB)
2.79.a_afm$4$(not in LMFDB)
2.79.abi_rf$6$(not in LMFDB)
2.79.aba_mp$6$(not in LMFDB)
2.79.ar_ic$6$(not in LMFDB)
2.79.an_dm$6$(not in LMFDB)
2.79.ae_acl$6$(not in LMFDB)
2.79.e_acl$6$(not in LMFDB)
2.79.n_dm$6$(not in LMFDB)
2.79.r_ic$6$(not in LMFDB)
2.79.ba_mp$6$(not in LMFDB)
2.79.bi_rf$6$(not in LMFDB)
2.79.a_l$12$(not in LMFDB)
2.79.a_fb$12$(not in LMFDB)