Invariants
Base field: | $\F_{199}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 50 x + 1021 x^{2} - 9950 x^{3} + 39601 x^{4}$ |
Frobenius angles: | $\pm0.114292864932$, $\pm0.184902183990$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1507904.2 |
Galois group: | $D_{4}$ |
Jacobians: | $15$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $30623$ | $1550228129$ | $62090456785700$ | $2459442447020867369$ | $97394093908836770643903$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $150$ | $39144$ | $7878900$ | $1568282724$ | $312080935750$ | $62103864122598$ | $12358664592937050$ | $2459374194781352964$ | $489415464141321686700$ | $97393677359702072427304$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 15 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=149x^6+76x^5+106x^4+150x^3+97x^2+77x+83$
- $y^2=97x^6+195x^5+180x^4+88x^3+61x^2+180x+114$
- $y^2=131x^6+152x^5+2x^4+93x^3+153x^2+193x+78$
- $y^2=66x^6+110x^5+28x^4+197x^3+46x^2+171x+80$
- $y^2=92x^6+26x^5+78x^4+135x^3+19x^2+144x+95$
- $y^2=120x^6+147x^5+171x^4+57x^3+198x^2+74x+169$
- $y^2=77x^6+17x^5+61x^4+70x^3+107x^2+93x+91$
- $y^2=101x^6+111x^5+184x^4+x^3+149x^2+186x+179$
- $y^2=189x^6+119x^5+31x^4+173x^3+101x^2+193x+78$
- $y^2=48x^6+142x^5+33x^4+8x^3+57x^2+55x+152$
- $y^2=180x^6+159x^5+160x^4+27x^3+188x^2+105x+170$
- $y^2=150x^6+66x^5+64x^4+189x^3+87x^2+138x+150$
- $y^2=62x^6+191x^5+152x^4+164x^3+166x^2+159x+152$
- $y^2=123x^6+67x^5+98x^4+127x^3+64x^2+76x+119$
- $y^2=83x^6+183x^5+7x^4+183x^3+173x^2+46x+88$
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 4.0.1507904.2. |
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 |
---|---|---|
2.199.by_bnh | $2$ | (not in LMFDB) |