Invariants
Base field: | $\F_{73}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 27 x + 324 x^{2} - 1971 x^{3} + 5329 x^{4}$ |
Frobenius angles: | $\pm0.135565313330$, $\pm0.266557904211$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.580312.1 |
Galois group: | $D_{4}$ |
Jacobians: | $18$ |
Isomorphism classes: | 18 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3656$ | $27975712$ | $151586724512$ | $806796370190464$ | $4297820565887899976$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $47$ | $5249$ | $389666$ | $28410081$ | $2073165527$ | $151334635502$ | $11047399015007$ | $806460094750849$ | $58871586930714002$ | $4297625833580755409$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=58x^6+20x^5+28x^4+68x^3+63x^2+14x+33$
- $y^2=3x^6+61x^5+x^4+64x^3+33x^2+23x+31$
- $y^2=19x^6+33x^5+30x^4+54x^3+45x^2+71x+26$
- $y^2=40x^6+40x^5+51x^4+70x^3+42x^2+70x+3$
- $y^2=68x^6+61x^5+51x^4+68x^3+14x^2+15x+47$
- $y^2=47x^6+32x^5+65x^4+51x^3+67x^2+29x+62$
- $y^2=28x^6+26x^5+16x^4+43x^3+4x^2+20x+47$
- $y^2=56x^6+8x^5+54x^4+36x^3+72x^2+72x+15$
- $y^2=66x^6+11x^5+10x^4+38x^3+27x^2+23x+22$
- $y^2=70x^6+29x^5+28x^4+28x^3+31x^2+16x+22$
- $y^2=53x^6+8x^5+67x^4+66x^3+58x^2+18x$
- $y^2=20x^6+14x^5+16x^4+63x^3+32x^2+69x+39$
- $y^2=19x^6+46x^5+62x^4+15x^3+48x^2+70x+65$
- $y^2=45x^6+3x^5+36x^4+35x^3+41x^2+43x+3$
- $y^2=71x^6+54x^5+12x^4+61x^3+17x^2+39x+22$
- $y^2=58x^6+10x^5+66x^4+55x^3+13x^2+15x+45$
- $y^2=66x^6+64x^5+39x^4+19x^3+60x^2+25x+50$
- $y^2=26x^6+20x^5+71x^4+55x^3+31x^2+39x+69$
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 4.0.580312.1. |
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 |
---|---|---|
2.73.bb_mm | $2$ | (not in LMFDB) |