Properties

Label 2.61.ax_ji
Base field $\F_{61}$
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:  $\F_{61}$
Dimension:  $2$
L-polynomial:  $( 1 - 15 x + 61 x^{2} )( 1 - 8 x + 61 x^{2} )$
  $1 - 23 x + 242 x^{2} - 1403 x^{3} + 3721 x^{4}$
Frobenius angles:  $\pm0.0900194921159$, $\pm0.328850104905$
Angle rank:  $2$ (numerical)
Jacobians:  $18$
Isomorphism classes:  150

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2538$ $13679820$ $51593316768$ $191719941336000$ $713313755584881858$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $39$ $3677$ $227304$ $13846753$ $844561779$ $51519979082$ $3142742179119$ $191707341720673$ $11694146486324904$ $713342914311311477$

Jacobians and polarizations

This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{61}$.

Endomorphism algebra over $\F_{61}$
The isogeny class factors as 1.61.ap $\times$ 1.61.ai 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
2.61.ah_c$2$(not in LMFDB)
2.61.h_c$2$(not in LMFDB)
2.61.x_ji$2$(not in LMFDB)