Invariants
Base field: | $\F_{59}$ |
Dimension: | $2$ |
L-polynomial: | $1 + 2 x + 44 x^{2} + 118 x^{3} + 3481 x^{4}$ |
Frobenius angles: | $\pm0.333832954610$, $\pm0.716465440511$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.3643200.2 |
Galois group: | $D_{4}$ |
Jacobians: | $144$ |
Isomorphism classes: | 312 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3646$ | $12418276$ | $42200338966$ | $146949185390800$ | $511087601918049046$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $62$ | $3566$ | $205478$ | $12127158$ | $714883522$ | $42179880206$ | $2488653088858$ | $146830433859358$ | $8662995974620862$ | $511116755238822206$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 144 curves (of which all are hyperelliptic):
- $y^2=15 x^6+4 x^5+40 x^4+53 x^3+11 x^2+56 x+12$
- $y^2=5 x^6+27 x^5+16 x^4+19 x^3+57 x^2+55 x+30$
- $y^2=38 x^6+39 x^5+34 x^4+36 x^3+24 x^2+8 x+7$
- $y^2=33 x^6+16 x^5+58 x^4+44 x^3+43 x^2+29 x+30$
- $y^2=47 x^6+26 x^5+34 x^4+55 x^3+24 x^2+8 x+18$
- $y^2=26 x^6+24 x^5+29 x^4+47 x^3+57 x^2+x+32$
- $y^2=19 x^6+33 x^5+47 x^4+56 x^3+57 x^2+29 x+10$
- $y^2=11 x^6+3 x^5+37 x^4+38 x^3+42 x^2+x+40$
- $y^2=53 x^6+22 x^5+37 x^4+47 x^3+14 x^2+33 x+14$
- $y^2=29 x^6+9 x^5+33 x^4+11 x^3+41 x^2+45 x+21$
- $y^2=27 x^5+47 x^4+28 x^3+21 x^2+45 x+13$
- $y^2=51 x^6+37 x^5+48 x^4+9 x^3+39 x^2+36 x+23$
- $y^2=5 x^6+19 x^5+30 x^3+9 x^2+40 x+57$
- $y^2=26 x^6+41 x^5+17 x^4+50 x^3+14 x^2+39 x+8$
- $y^2=47 x^6+47 x^5+33 x^4+56 x^3+53 x^2+43 x+7$
- $y^2=33 x^6+2 x^5+25 x^4+17 x^3+13 x^2+40 x+15$
- $y^2=53 x^6+45 x^5+14 x^3+45 x^2+43 x+35$
- $y^2=7 x^6+44 x^5+58 x^4+50 x^3+45 x^2+57 x$
- $y^2=53 x^6+31 x^5+7 x^4+50 x^3+54 x^2+3 x+16$
- $y^2=48 x^6+24 x^5+33 x^4+39 x^3+27 x^2+24 x+50$
- and 124 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59}$.
Endomorphism algebra over $\F_{59}$The endomorphism algebra of this simple isogeny class is 4.0.3643200.2. |
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 |
---|---|---|
2.59.ac_bs | $2$ | (not in LMFDB) |