Invariants
Base field: | $\F_{41}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 18 x + 149 x^{2} - 738 x^{3} + 1681 x^{4}$ |
Frobenius angles: | $\pm0.0319832925581$, $\pm0.365316625891$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{14})\) |
Galois group: | $C_2^2$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
This isogeny class is simple but not geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1075$ | $2781025$ | $4749990700$ | $7977339621225$ | $13418341001726875$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $24$ | $1656$ | $68922$ | $2823076$ | $115818924$ | $4749877158$ | $194753758524$ | $7984926792196$ | $327381934393962$ | $13422659078648376$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=38x^6+16x^5+15x^4+31x^3+14x^2+37x+11$
- $y^2=28x^6+14x^5+26x^4+16x^3+7x^2+31x+28$
- $y^2=27x^6+8x^5+37x^4+8x^3+3x^2+5x+24$
- $y^2=13x^6+35x^5+37x^4+29x^3+16x^2+2x+13$
- $y^2=37x^6+19x^5+20x^4+9x^3+15x^2+24x+14$
- $y^2=34x^6+36x^5+36x^4+33x^3+15x^2+34x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{41^{6}}$.
Endomorphism algebra over $\F_{41}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{14})\). |
The base change of $A$ to $\F_{41^{6}}$ is 1.4750104241.aglza 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$ |
- Endomorphism algebra over $\F_{41^{2}}$
The base change of $A$ to $\F_{41^{2}}$ is the simple isogeny class 2.1681.aba_abmr and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{14})\). - Endomorphism algebra over $\F_{41^{3}}$
The base change of $A$ to $\F_{41^{3}}$ is the simple isogeny class 2.68921.a_aglza and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{14})\).
Base change
This is a primitive isogeny class.