Invariants
Base field: | $\F_{193}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 50 x + 1006 x^{2} - 9650 x^{3} + 37249 x^{4}$ |
Frobenius angles: | $\pm0.0633713900940$, $\pm0.194366849898$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.191600.1 |
Galois group: | $D_{4}$ |
Jacobians: | $24$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $28556$ | $1369431536$ | $51660622163084$ | $1925129784714135296$ | $71709030129753846498636$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $144$ | $36762$ | $7186008$ | $1387492926$ | $267785651944$ | $51682547728218$ | $9974730375523008$ | $1925122952503205118$ | $371548729894612587744$ | $71708904872970152386522$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=189x^6+134x^5+71x^4+114x^3+116x^2+182x+103$
- $y^2=141x^6+152x^5+132x^4+130x^3+36x^2+3x+64$
- $y^2=37x^6+35x^5+171x^4+191x^3+188x^2+187x+122$
- $y^2=82x^6+68x^5+125x^4+46x^3+88x^2+169x+148$
- $y^2=147x^6+128x^5+51x^4+22x^3+123x^2+31x+44$
- $y^2=102x^6+149x^5+146x^4+33x^3+6x^2+128x+151$
- $y^2=31x^6+98x^5+95x^4+177x^3+68x^2+167x+56$
- $y^2=40x^6+113x^5+91x^4+66x^3+136x^2+157x+171$
- $y^2=177x^6+150x^5+88x^4+118x^3+172x^2+166x+1$
- $y^2=69x^5+45x^4+149x^3+65x^2+61x+111$
- $y^2=180x^6+28x^5+18x^4+122x^3+53x^2+166x+178$
- $y^2=113x^6+52x^5+22x^4+184x^3+123x^2+181x+15$
- $y^2=105x^6+135x^5+3x^4+8x^3+84x^2+97x+57$
- $y^2=8x^6+95x^5+9x^4+132x^3+63x^2+105x+163$
- $y^2=161x^6+163x^5+92x^4+46x^3+97x^2+97x+165$
- $y^2=87x^6+129x^5+87x^4+26x^3+17x^2+22x+189$
- $y^2=142x^6+18x^5+43x^4+181x^3+68x^2+32x+133$
- $y^2=46x^6+151x^5+174x^4+114x^3+145x^2+16x+179$
- $y^2=182x^6+79x^5+82x^4+18x^3+98x^2+50x+127$
- $y^2=104x^6+159x^5+124x^4+173x^3+168x^2+63x+60$
- $y^2=99x^6+41x^5+6x^4+138x^3+158x^2+5x+60$
- $y^2=188x^6+48x^5+164x^4+144x^3+101x^2+66x+156$
- $y^2=46x^6+97x^5+182x^4+153x^3+46x^2+72x+178$
- $y^2=74x^6+171x^5+17x^4+75x^3+82x^2+163x+51$
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 4.0.191600.1. |
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_bms | $2$ | (not in LMFDB) |