Properties

Label 2.71.abd_nm
Base field $\F_{71}$
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:  $\F_{71}$
Dimension:  $2$
L-polynomial:  $( 1 - 16 x + 71 x^{2} )( 1 - 13 x + 71 x^{2} )$
  $1 - 29 x + 350 x^{2} - 2059 x^{3} + 5041 x^{4}$
Frobenius angles:  $\pm0.101666819831$, $\pm0.219552767034$
Angle rank:  $2$ (numerical)
Jacobians:  $8$
Isomorphism classes:  20

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $3304$ $24713920$ $128059088416$ $645917180634880$ $3255393668484748504$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $43$ $4901$ $357796$ $25418121$ $1804312553$ $128100915038$ $9095123183063$ $645753536922481$ $45848500704490636$ $3255243551664577901$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{71}$.

Endomorphism algebra over $\F_{71}$
The isogeny class factors as 1.71.aq $\times$ 1.71.an 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 is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.71.ad_aco$2$(not in LMFDB)
2.71.d_aco$2$(not in LMFDB)
2.71.bd_nm$2$(not in LMFDB)