Invariants
Base field: | $\F_{23}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 12 x + 72 x^{2} - 276 x^{3} + 529 x^{4}$ |
Frobenius angles: | $\pm0.0956038575290$, $\pm0.404396142471$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(i, \sqrt{10})\) |
Galois group: | $C_2^2$ |
Jacobians: | $7$ |
Isomorphism classes: | 14 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $314$ | $279460$ | $148474586$ | $78097891600$ | $41391692364794$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $12$ | $530$ | $12204$ | $279078$ | $6430932$ | $148035890$ | $3404985204$ | $78311812798$ | $1801153952172$ | $41426511213650$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=13x^6+5x^5+21x^4+21x^2+18x+13$
- $y^2=10x^6+13x^5+19x^4+5x^3+19x^2+11x+7$
- $y^2=9x^6+14x^5+13x^4+15x^3+9x^2+16x+21$
- $y^2=4x^6+17x^5+17x^4+7x^2+12x+9$
- $y^2=9x^6+6x^5+20x^4+16x^3+18x^2+18x+14$
- $y^2=7x^6+19x^5+10x^4+6x^3+10x^2+9x+18$
- $y^2=16x^6+17x^5+20x^3+16x^2+6x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23^{4}}$.
Endomorphism algebra over $\F_{23}$The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{10})\). |
The base change of $A$ to $\F_{23^{4}}$ is 1.279841.aos 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$ |
- Endomorphism algebra over $\F_{23^{2}}$
The base change of $A$ to $\F_{23^{2}}$ is the simple isogeny class 2.529.a_aos and its endomorphism algebra is \(\Q(i, \sqrt{10})\).
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 |
---|---|---|
2.23.m_cu | $2$ | (not in LMFDB) |
2.23.a_aba | $8$ | (not in LMFDB) |
2.23.a_ba | $8$ | (not in LMFDB) |