Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 10 x + 50 x^{2} - 190 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.0511346671405$, $\pm0.448865332859$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(i, \sqrt{13})\) |
Galois group: | $C_2^2$ |
Jacobians: | $8$ |
Isomorphism classes: | 11 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $212$ | $129744$ | $46568132$ | $16833505536$ | $6120057514052$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $362$ | $6790$ | $129166$ | $2471650$ | $47045882$ | $893899870$ | $16983416158$ | $322686428890$ | $6131066257802$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=17x^6+16x^5+4x^4+10x^3+13x^2+3x+12$
- $y^2=10x^6+15x^5+2x^4+13x^3+16x^2+12x+8$
- $y^2=9x^6+14x^5+x^4+11x^3+x^2+2x+2$
- $y^2=12x^5+2x^4+6x^3+x^2+2x+6$
- $y^2=3x^6+18x^5+17x^4+9x^3+8x^2+10x+13$
- $y^2=8x^5+3x^4+11x^3+3x^2+9x+16$
- $y^2=13x^6+9x^5+15x^4+12x^3+15x^2+4x+9$
- $y^2=5x^5+16x^4+16x^2+14x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{4}}$.
Endomorphism algebra over $\F_{19}$The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{13})\). |
The base change of $A$ to $\F_{19^{4}}$ is 1.130321.awg 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$ |
- Endomorphism algebra over $\F_{19^{2}}$
The base change of $A$ to $\F_{19^{2}}$ is the simple isogeny class 2.361.a_awg and its endomorphism algebra is \(\Q(i, \sqrt{13})\).
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.k_by | $2$ | (not in LMFDB) |
2.19.a_am | $8$ | (not in LMFDB) |
2.19.a_m | $8$ | (not in LMFDB) |