Invariants
Base field: | $\F_{73}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 30 x + 369 x^{2} - 2190 x^{3} + 5329 x^{4}$ |
Frobenius angles: | $\pm0.0896851289724$, $\pm0.207445605396$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.155200.2 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3479$ | $27550201$ | $151194570716$ | $806590568251321$ | $4297782867694392839$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $44$ | $5168$ | $388658$ | $28402836$ | $2073147344$ | $151334878502$ | $11047402113008$ | $806460100456228$ | $58871586663652034$ | $4297625829600674528$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=33x^6+13x^5+41x^4+28x^3+49x^2+46x+42$
- $y^2=61x^6+52x^5+32x^4+12x^3+38x^2+46x+68$
- $y^2=7x^6+51x^5+67x^4+68x^3+50x^2+58x+68$
- $y^2=40x^6+42x^5+60x^4+65x^3+9x^2+4x+21$
- $y^2=45x^6+x^5+20x^4+59x^3+6x^2+36x+63$
- $y^2=43x^6+28x^5+43x^4+71x^3+43x^2+19x+30$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{73}$.
Endomorphism algebra over $\F_{73}$The endomorphism algebra of this simple isogeny class is 4.0.155200.2. |
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 |
---|---|---|
2.73.be_of | $2$ | (not in LMFDB) |