Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 49 x + 989 x^{2} - 9751 x^{3} + 39601 x^{4}$ |
Frobenius angles: | $\pm0.0696172273604$, $\pm0.224914406157$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1912493.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30791$ | $1551589281$ | $62091872695277$ | $2459412645659510013$ | $97393839515101513312016$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $151$ | $39179$ | $7879081$ | $1568263723$ | $312080120596$ | $62103843747191$ | $12358664218810207$ | $2459374189577677699$ | $489415464092084717173$ | $97393677359613282430814$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=97x^6+60x^5+132x^4+133x^3+23x^2+51x+112$
- $y^2=191x^6+140x^5+119x^4+15x^3+85x^2+35x+73$
- $y^2=85x^6+182x^5+100x^3+197x^2+143x+146$
- $y^2=131x^6+92x^5+165x^4+130x^3+131x^2+35x+136$
- $y^2=177x^6+180x^5+126x^4+50x^3+26x^2+67x+179$
- $y^2=177x^6+24x^5+133x^4+172x^3+28x^2+152x+196$
- $y^2=49x^6+122x^5+140x^4+134x^3+56x^2+117x+161$
- $y^2=108x^6+30x^5+151x^4+42x^3+4x^2+39x+166$
- $y^2=22x^6+188x^5+18x^4+149x^3+14x^2+21x+23$
- $y^2=88x^6+160x^5+75x^4+68x^3+8x^2+104x+138$
- $y^2=68x^6+186x^5+80x^4+11x^3+3x^2+50x+12$
- $y^2=160x^6+63x^5+43x^4+17x^3+107x^2+13x+124$
- $y^2=164x^6+148x^5+135x^4+147x^3+182x^2+163x+124$
- $y^2=97x^6+32x^5+136x^4+144x^3+88x^2+147x+139$
- $y^2=39x^6+109x^5+77x^4+47x^3+118x^2+67x+48$
- $y^2=128x^6+178x^5+28x^4+46x^3+148x^2+66x+103$
- $y^2=166x^6+95x^5+183x^4+64x^3+81x^2+x+34$
- $y^2=117x^6+48x^5+91x^4+38x^3+185x^2+13x+181$
- $y^2=37x^6+66x^5+129x^4+98x^3+121x^2+159x+114$
- $y^2=184x^6+94x^5+159x^4+107x^3+46x^2+91x+45$
- $y^2=86x^6+72x^5+157x^4+7x^3+56x^2+27x+147$
- $y^2=115x^6+196x^5+83x^4+139x^3+43x^2+127x+77$
- $y^2=133x^6+73x^5+32x^4+45x^3+14x^2+71x+139$
- $y^2=42x^6+32x^5+82x^4+27x^3+111x^2+13x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{199}$.
Endomorphism algebra over $\F_{199}$The endomorphism algebra of this simple isogeny class is 4.0.1912493.1. |
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 |
---|---|---|
2.199.bx_bmb | $2$ | (not in LMFDB) |