Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 45 x + 840 x^{2} - 7605 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.106377140183$, $\pm0.212100084678$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1357144.1 |
Galois group: | $D_{4}$ |
Jacobians: | $42$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21752$ | $805955104$ | $23295483984512$ | $665447215543159936$ | $19005074541062408349752$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $125$ | $28217$ | $4826270$ | $815768241$ | $137859295325$ | $23298095509742$ | $3937376478961925$ | $665416609693547233$ | $112455406952630375150$ | $19004963774894966827577$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 42 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(10a+3)x^6+(2a+8)x^5+(10a+5)x^4+(10a+6)x^3+(2a+6)x^2+(10a+11)x+11a+1$
- $y^2=(9a+8)x^6+(6a+7)x^5+(9a+6)x^4+(4a+8)x^3+(a+1)x^2+(2a+7)x+11a+2$
- $y^2=(2a+5)x^6+(11a+2)x^5+(10a+8)x^4+(a+9)x^3+(9a+6)x^2+(9a+9)x+4a+5$
- $y^2=(4a+11)x^6+(9a+6)x^5+(12a+4)x^4+(2a+9)x^3+(12a+11)x^2+(2a+10)x+5a+7$
- $y^2=(12a+8)x^6+(7a+9)x^5+(10a+6)x^4+(12a+1)x^3+(3a+11)x^2+(7a+12)x+12a$
- $y^2=(8a+2)x^6+(12a+5)x^5+(8a+5)x^4+(a+12)x^3+(a+12)x^2+(5a+7)x+9a+2$
- $y^2=(8a+3)x^6+(11a+8)x^5+(a+2)x^4+5ax^3+8ax^2+(9a+6)x+2a+10$
- $y^2=(7a+6)x^6+(11a+5)x^5+(6a+4)x^4+(6a+4)x^2+x+9a+8$
- $y^2=(10a+8)x^6+(7a+10)x^5+(4a+10)x^4+(4a+6)x^3+(2a+6)x^2+3ax+6a+8$
- $y^2=(9a+3)x^6+11ax^5+(3a+2)x^4+(5a+2)x^3+(a+7)x^2+(2a+9)x+3$
- $y^2=(3a+2)x^6+(5a+10)x^5+(2a+8)x^4+(8a+3)x^3+(6a+1)x^2+(4a+11)x+3a+11$
- $y^2=(a+7)x^6+(10a+10)x^5+(10a+7)x^4+(4a+9)x^3+(10a+8)x^2+(11a+9)x+10a+8$
- $y^2=(6a+12)x^6+(9a+1)x^5+9ax^4+(10a+8)x^3+(6a+6)x^2+(2a+4)x+9a+11$
- $y^2=(9a+4)x^6+(6a+5)x^5+(6a+3)x^4+2x^3+(8a+12)x^2+12ax+2a+11$
- $y^2=(5a+5)x^6+(7a+9)x^5+(10a+2)x^3+(9a+4)x^2+12x+6a$
- $y^2=7ax^6+(4a+1)x^5+(a+7)x^4+(6a+1)x^3+(2a+10)x^2+6ax+10a+7$
- $y^2=(5a+4)x^6+(8a+10)x^5+10x^4+11ax^3+(2a+3)x^2+(8a+10)x+7a+6$
- $y^2=(8a+5)x^6+(12a+3)x^5+(3a+2)x^4+(11a+10)x^3+(10a+12)x^2+(7a+7)x+10a+8$
- $y^2=8ax^6+(10a+8)x^5+(6a+9)x^4+(3a+1)x^3+(10a+11)x^2+(12a+1)x+11a+6$
- $y^2=(4a+9)x^6+(12a+2)x^5+(2a+4)x^4+(12a+4)x^3+(3a+2)x^2+(9a+12)x+11a+3$
- and 22 more
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.1357144.1. |
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 |
---|---|---|
2.169.bt_bgi | $2$ | (not in LMFDB) |