Invariants
Base field: | $\F_{73}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 30 x + 368 x^{2} - 2190 x^{3} + 5329 x^{4}$ |
Frobenius angles: | $\pm0.0650840807172$, $\pm0.217019455223$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.201024.4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3478$ | $27538804$ | $151159364806$ | $806530164455376$ | $4297709478118732918$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $44$ | $5166$ | $388568$ | $28400710$ | $2073111944$ | $151334417022$ | $11047397195228$ | $806460057203134$ | $58871586357401564$ | $4297625828028149886$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=58x^6+65x^5+29x^4+71x^3+70x^2+49x+45$
- $y^2=10x^5+68x^4+12x^3+42x^2+40x+30$
- $y^2=50x^6+5x^5+13x^4+30x^3+41x^2+14x+5$
- $y^2=52x^6+17x^5+72x^4+62x^3+62x^2+37x+3$
- $y^2=71x^6+40x^5+13x^4+56x^3+50x^2+10x+40$
- $y^2=24x^6+68x^5+35x^4+25x^3+23x^2+23x$
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 4.0.201024.4. |
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 |
---|---|---|
2.73.be_oe | $2$ | (not in LMFDB) |