Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 12 x + 41 x^{2} )( 1 - 6 x + 41 x^{2} )$ |
$1 - 18 x + 154 x^{2} - 738 x^{3} + 1681 x^{4}$ | |
Frobenius angles: | $\pm0.113551764296$, $\pm0.344786929280$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $46$ |
Isomorphism classes: | 280 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1080$ | $2799360$ | $4768719480$ | $7987089162240$ | $13421572979067000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $24$ | $1666$ | $69192$ | $2826526$ | $115846824$ | $4750043938$ | $194754742584$ | $7984934747326$ | $327382004801592$ | $13422659558959426$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 46 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+33x^5+15x^4+6x^3+15x^2+33x+8$
- $y^2=14x^6+25x^5+7x^4+25x^3+24x^2+28x+6$
- $y^2=35x^6+23x^5+38x^4+21x^3+38x^2+23x+35$
- $y^2=3x^6+5x^5+20x^3+11x^2+25x+19$
- $y^2=35x^6+38x^5+12x^4+7x^3+39x^2+25x+12$
- $y^2=34x^6+32x^5+15x^4+4x^3+15x^2+32x+34$
- $y^2=30x^5+24x^4+26x^3+9x^2+18x+29$
- $y^2=14x^6+20x^5+4x^4+30x^3+19x^2+28x+3$
- $y^2=16x^6+35x^5+5x^4+19x^3+5x^2+12x+25$
- $y^2=31x^6+19x^5+32x^4+33x^3+39x^2+11x+2$
- $y^2=7x^6+40x^5+16x^4+11x^3+16x^2+40x+7$
- $y^2=22x^6+38x^5+25x^4+35x^3+25x^2+36x+15$
- $y^2=9x^6+17x^5+20x^4+20x^3+17x^2+30x+37$
- $y^2=3x^6+7x^5+28x^4+21x^3+28x^2+7x+3$
- $y^2=22x^6+31x^5+31x^4+19x^3+12x^2+23x+14$
- $y^2=3x^6+x^5+27x^4+14x^3+24x^2+18x+7$
- $y^2=8x^6+12x^5+29x^4+3x^3+29x^2+29x+19$
- $y^2=9x^6+16x^4+23x^3+8x^2+37$
- $y^2=37x^6+6x^5+33x^4+21x^3+2x^2+22x+13$
- $y^2=6x^6+25x^5+19x^4+7x^3+29x^2+18x+38$
- and 26 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41}$.
Endomorphism algebra over $\F_{41}$The isogeny class factors as 1.41.am $\times$ 1.41.ag 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.
Twist | Extension degree | Common base change |
---|---|---|
2.41.ag_k | $2$ | (not in LMFDB) |
2.41.g_k | $2$ | (not in LMFDB) |
2.41.s_fy | $2$ | (not in LMFDB) |