Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 12 x + 68 x^{2} - 228 x^{3} + 361 x^{4}$ |
Frobenius angles: | $\pm0.0791769653548$, $\pm0.366480294854$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.168192.4 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $190$ | $127300$ | $47287390$ | $16942611600$ | $6121833155950$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $354$ | $6896$ | $130006$ | $2472368$ | $47033682$ | $893894072$ | $16983923806$ | $322689074024$ | $6131067736674$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+5x^5+3x^4+8x^3+6x^2+13x+7$
- $y^2=18x^6+4x^5+x^4+8x^3+9x^2+10x+11$
- $y^2=4x^6+18x^5+4x^4+13x^3+4x^2+2x+5$
- $y^2=2x^6+14x^5+13x^4+11x^3+3x+8$
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 4.0.168192.4. |
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_cq | $2$ | (not in LMFDB) |