Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 12 x + 41 x^{2} )( 1 - 9 x + 41 x^{2} )$ |
$1 - 21 x + 190 x^{2} - 861 x^{3} + 1681 x^{4}$ | |
Frobenius angles: | $\pm0.113551764296$, $\pm0.251940962052$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 30 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $990$ | $2726460$ | $4758831000$ | $7993064629440$ | $13424895517794750$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $21$ | $1621$ | $69048$ | $2828641$ | $115875501$ | $4750173538$ | $194754329301$ | $7984925003521$ | $327381946348968$ | $13422659535300901$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=11x^6+30x^5+13x^4+16x^3+19x^2+15x+29$
- $y^2=35x^6+40x^5+36x^4+9x^3+15x^2+22$
- $y^2=13x^6+34x^5+30x^4+36x^3+14x^2+35x+1$
- $y^2=22x^6+15x^5+11x^4+35x^3+38x^2+30x+40$
- $y^2=11x^6+18x^5+11x^4+3x^3+36x^2+4x+14$
- $y^2=35x^6+31x^5+4x^4+35x^3+3x^2+2x+37$
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.aj 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.ad_aba | $2$ | (not in LMFDB) |
2.41.d_aba | $2$ | (not in LMFDB) |
2.41.v_hi | $2$ | (not in LMFDB) |