Invariants
Base field: | $\F_{23}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 23 x^{2} )( 1 - 3 x + 23 x^{2} )$ |
$1 - 12 x + 73 x^{2} - 276 x^{3} + 529 x^{4}$ | |
Frobenius angles: | $\pm0.112386341891$, $\pm0.398742550628$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $21$ |
Isomorphism classes: | 60 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $315$ | $280665$ | $148916880$ | $78177832425$ | $41400181590075$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $12$ | $532$ | $12240$ | $279364$ | $6432252$ | $148040494$ | $3405009396$ | $78311980036$ | $1801154605680$ | $41426510261332$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 21 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+21x^5+19x^4+x^3+14x^2+10x+12$
- $y^2=2x^6+14x^5+22x^4+9x^3+22x^2+2x+19$
- $y^2=22x^6+7x^5+20x^4+2x^3+11x^2+19x+15$
- $y^2=2x^6+12x^5+5x^4+17x^3+19x^2+4x+1$
- $y^2=17x^6+7x^5+22x^4+15x^3+20x^2+17x+22$
- $y^2=12x^6+13x^5+9x^4+7x^3+8x^2+6x+19$
- $y^2=x^6+22x^5+14x^4+15x^3+14x^2+22x+1$
- $y^2=20x^6+3x^5+3x^4+5x^3+4x^2+11x+21$
- $y^2=22x^6+13x^5+5x^4+9x^3+5x^2+13x+22$
- $y^2=4x^6+6x^5+21x^4+6x^3+21x^2+14x+17$
- $y^2=10x^6+15x^5+13x^4+22x^3+9x^2+9x+12$
- $y^2=22x^6+19x^5+5x^4+4x^3+19x^2+17x+4$
- $y^2=7x^6+20x^5+11x^4+15x^3+22x^2+13x+19$
- $y^2=2x^6+12x^5+8x^4+8x^3+6x^2+x+3$
- $y^2=10x^6+11x^5+15x^4+x^3+11x^2+18x+2$
- $y^2=19x^6+21x^5+8x^4+12x^3+13x^2+17x+15$
- $y^2=11x^6+15x^5+3x^4+11x^3+20x^2+5x+11$
- $y^2=20x^6+5x^4+11x^3+11x^2+17$
- $y^2=22x^6+x^5+16x^4+16x^3+4x^2+16x+19$
- $y^2=20x^6+x^5+2x^4+5x^3+5x^2+18x+17$
- $y^2=x^6+12x^5+19x^4+9x^3+17x^2+4x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{23}$.
Endomorphism algebra over $\F_{23}$The isogeny class factors as 1.23.aj $\times$ 1.23.ad 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 |
---|---|---|
2.23.ag_t | $2$ | (not in LMFDB) |
2.23.g_t | $2$ | (not in LMFDB) |
2.23.m_cv | $2$ | (not in LMFDB) |