Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 18 x + 162 x^{2} + 1062 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.560815078026$, $\pm0.939184921974$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{37})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $20$ |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4724$ | $12112336$ | $42236022692$ | $146708683376896$ | $511170045106071524$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $78$ | $3482$ | $205650$ | $12107310$ | $714998838$ | $42180533642$ | $2488649145018$ | $146830435552414$ | $8662995975170430$ | $511116753300641402$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=46 x^6+48 x^5+22 x^4+29 x^3+56 x^2+11 x+1$
- $y^2=19 x^6+27 x^5+22 x^4+57 x^3+44 x^2+5 x+39$
- $y^2=26 x^6+26 x^5+17 x^4+35 x^3+32 x+27$
- $y^2=49 x^6+47 x^5+48 x^4+26 x^3+22 x^2+7 x+14$
- $y^2=46 x^6+47 x^5+13 x^4+41 x^3+x^2+27 x+41$
- $y^2=36 x^6+38 x^5+2 x^4+55 x^3+38 x^2+46$
- $y^2=29 x^6+3 x^5+13 x^4+10 x^3+47 x^2+33 x+19$
- $y^2=44 x^6+5 x^5+6 x^4+41 x^3+27 x^2+4 x+14$
- $y^2=7 x^6+48 x^5+26 x^4+8 x^3+6 x^2+45 x+22$
- $y^2=57 x^6+28 x^5+42 x^4+41 x^3+3 x^2+37 x+48$
- $y^2=29 x^6+54 x^5+19 x^4+35 x^3+50 x^2+6 x+26$
- $y^2=57 x^6+37 x^5+10 x^4+41 x^3+14 x^2+2 x+4$
- $y^2=2 x^6+8 x^5+57 x^4+32 x^3+24 x^2+42 x+20$
- $y^2=21 x^6+42 x^5+45 x^4+48 x^3+51 x^2+58 x+19$
- $y^2=10 x^6+5 x^5+19 x^4+49 x^3+45 x^2+49 x+2$
- $y^2=26 x^6+3 x^5+55 x^4+55 x^2+56 x+26$
- $y^2=20 x^6+11 x^5+x^4+9 x^3+22 x^2+26 x+31$
- $y^2=21 x^6+14 x^5+28 x^4+34 x^3+40 x^2+32 x+53$
- $y^2=36 x^6+45 x^5+51 x^4+9 x^3+7 x^2+46 x+21$
- $y^2=40 x^6+56 x^5+16 x^4+48 x^3+52 x^2+28 x+16$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59^{4}}$.
Endomorphism algebra over $\F_{59}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{37})\). |
| The base change of $A$ to $\F_{59^{4}}$ is 1.12117361.ahli 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-37}) \)$)$ |
- Endomorphism algebra over $\F_{59^{2}}$
The base change of $A$ to $\F_{59^{2}}$ is the simple isogeny class 2.3481.a_ahli and its endomorphism algebra is \(\Q(i, \sqrt{37})\).
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.as_gg | $2$ | (not in LMFDB) |
| 2.59.a_abs | $8$ | (not in LMFDB) |
| 2.59.a_bs | $8$ | (not in LMFDB) |