Invariants
Base field: | $\F_{13^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 45 x + 839 x^{2} - 7605 x^{3} + 28561 x^{4}$ |
Frobenius angles: | $\pm0.0974399185523$, $\pm0.216609910606$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.7245189.1 |
Galois group: | $D_{4}$ |
Jacobians: | $12$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21751$ | $805896301$ | $23294831363239$ | $665443347009001029$ | $19005058566634093568176$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $125$ | $28215$ | $4826135$ | $815763499$ | $137859179450$ | $23298093341871$ | $3937376447223155$ | $665416609348740499$ | $112455406950659009705$ | $19004963774916601817550$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(10a+8)x^6+(6a+12)x^5+(7a+12)x^4+(7a+6)x^3+(4a+11)x^2+(a+8)x+3a+8$
- $y^2=(9a+4)x^6+x^5+5x^4+(10a+9)x^3+(11a+4)x^2+(2a+7)x+5a+4$
- $y^2=(6a+10)x^6+(5a+7)x^5+(8a+9)x^4+(8a+2)x^3+(9a+3)x^2+(11a+2)x+a+12$
- $y^2=(4a+2)x^6+(a+1)x^5+(11a+12)x^4+(a+4)x^3+(9a+6)x^2+10ax+6a+7$
- $y^2=(7a+10)x^6+6x^5+(5a+6)x^4+(2a+1)x^3+7x^2+(11a+5)x+7a+1$
- $y^2=(10a+8)x^6+(4a+10)x^5+(7a+6)x^4+4ax^3+(2a+5)x^2+(a+3)x+1$
- $y^2=(8a+9)x^6+(4a+3)x^5+(7a+8)x^4+(9a+4)x^3+(8a+8)x^2+(3a+11)x+2a+10$
- $y^2=(6a+12)x^6+(12a+2)x^5+(9a+6)x^4+(8a+5)x^3+(a+10)x^2+(4a+1)x+5a$
- $y^2=(11a+8)x^6+(10a+8)x^5+3x^4+(3a+2)x^3+(12a+2)x^2+(7a+10)x+6a+4$
- $y^2=(9a+3)x^6+10ax^5+8ax^4+(11a+2)x^3+(a+4)x^2+(12a+8)x+4a+7$
- $y^2=(9a+12)x^6+3ax^5+(9a+8)x^4+(9a+4)x^3+(8a+10)x^2+(7a+5)x+10a+2$
- $y^2=(5a+1)x^6+(6a+1)x^5+(7a+10)x^4+(8a+2)x^3+(8a+5)x^2+(2a+4)x+12a+1$
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 4.0.7245189.1. |
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_bgh | $2$ | (not in LMFDB) |