Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 50 x + 1009 x^{2} - 9650 x^{3} + 37249 x^{4}$ |
Frobenius angles: | $\pm0.100399071344$, $\pm0.177282845889$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1025600.4 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28559$ | $1369661081$ | $51663861696956$ | $1925154636563717561$ | $71709166299106868350239$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $144$ | $36768$ | $7186458$ | $1387510836$ | $267786160444$ | $51682559186262$ | $9974730590014908$ | $1925122955884992228$ | $371548729938760070394$ | $71708904873409596791728$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=100x^6+187x^5+59x^4+7x^3+41x^2+171x+109$
- $y^2=104x^6+21x^5+182x^4+118x^3+29x^2+54x+84$
- $y^2=135x^6+150x^5+161x^4+66x^3+28x^2+29x+53$
- $y^2=137x^6+134x^5+176x^4+45x^3+110x^2+146x+74$
- $y^2=137x^6+167x^5+167x^4+30x^3+132x^2+24x+130$
- $y^2=102x^6+41x^5+158x^4+137x^3+18x^2+107x+178$
- $y^2=10x^6+56x^5+22x^4+51x^3+189x^2+4x+98$
- $y^2=133x^6+39x^5+48x^4+149x^3+61x^2+103x+64$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{193}$.
Endomorphism algebra over $\F_{193}$The endomorphism algebra of this simple isogeny class is 4.0.1025600.4. |
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 |
---|---|---|
2.193.by_bmv | $2$ | (not in LMFDB) |