Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 6 x + 19 x^{2} )( 1 - 4 x + 19 x^{2} )$ |
$1 - 10 x + 62 x^{2} - 190 x^{3} + 361 x^{4}$ | |
Frobenius angles: | $\pm0.258380448083$, $\pm0.348268167089$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
Isomorphism classes: | 32 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $224$ | $139776$ | $49069664$ | $17108582400$ | $6130448043104$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $386$ | $7150$ | $131278$ | $2475850$ | $47030546$ | $893810830$ | $16983512158$ | $322688141770$ | $6131067815906$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+17x^5+11x^3+17x+4$
- $y^2=4x^6+14x^5+x^4+17x^2+18x+6$
- $y^2=9x^6+18x^5+5x^4+17x^3+5x^2+18x+9$
- $y^2=x^6+17x^5+7x^4+3x^3+17x^2+x+11$
- $y^2=14x^6+15x^5+10x^4+11x^3+10x^2+15x+14$
- $y^2=8x^6+8x^5+x^4+5x^3+4x^2+14x+18$
- $y^2=10x^6+18x^5+10x^4+14x^3+10x^2+18x+10$
- $y^2=13x^6+4x^5+17x^4+x^2+x+15$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$The isogeny class factors as 1.19.ag $\times$ 1.19.ae 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 |
---|---|---|
2.19.ac_o | $2$ | (not in LMFDB) |
2.19.c_o | $2$ | (not in LMFDB) |
2.19.k_ck | $2$ | (not in LMFDB) |