Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 12 x + 71 x^{2} - 228 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.152835520717$, $\pm0.337154730080$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.134928.2 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
This isogeny class is simple and 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $193$ | $129889$ | $48043876$ | $17059231593$ | $6132523129513$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $360$ | $7004$ | $130900$ | $2476688$ | $47045454$ | $893904656$ | $16983913828$ | $322689260324$ | $6131067761880$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=13x^6+15x^5+10x^4+x^3+12x^2+2$
- $y^2=9x^6+10x^5+12x^4+12x^3+16x^2+3x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$The endomorphism algebra of this simple isogeny class is 4.0.134928.2. |
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.m_ct | $2$ | (not in LMFDB) |