Invariants
Base field: | $\F_{61}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 14 x + 61 x^{2} )( 1 - 12 x + 61 x^{2} )$ |
$1 - 26 x + 290 x^{2} - 1586 x^{3} + 3721 x^{4}$ | |
Frobenius angles: | $\pm0.146275019398$, $\pm0.221142061624$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $10$ |
Isomorphism classes: | 40 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2400$ | $13497600$ | $51585660000$ | $191830914662400$ | $713421947728860000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $36$ | $3626$ | $227268$ | $13854766$ | $844689876$ | $51521030138$ | $3142745838996$ | $191707316101726$ | $11694145994559108$ | $713342910530017226$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=30x^6+10x^5+51x^4+13x^3+51x^2+10x+30$
- $y^2=46x^6+47x^4+14x^3+12x^2+5$
- $y^2=11x^6+49x^5+52x^4+37x^3+52x^2+49x+11$
- $y^2=26x^6+4x^5+41x^4+22x^3+41x^2+4x+26$
- $y^2=14x^6+23x^5+11x^4+47x^3+8x^2+53x+36$
- $y^2=51x^6+24x^5+39x^4+24x^3+39x^2+24x+51$
- $y^2=33x^6+6x^5+7x^4+21x^3+6x^2+43x+11$
- $y^2=5x^6+25x^5+58x^4+25x^3+22x^2+16x+5$
- $y^2=4x^6+21x^5+33x^4+20x^3+43x^2+37x+12$
- $y^2=44x^6+52x^5+60x^4+31x^3+60x^2+52x+44$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{61}$.
Endomorphism algebra over $\F_{61}$The isogeny class factors as 1.61.ao $\times$ 1.61.am 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.