Normalized defining polynomial
\( x^{6} - 3x^{5} - 84x^{4} + 171x^{3} + 2109x^{2} - 2352x - 15391 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(8496685269\) \(\medspace = 3^{8}\cdot 109^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.17\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $3^{4/3}109^{1/2}\approx 45.17258272906164$ | ||
Ramified primes: | \(3\), \(109\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{109}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(981=3^{2}\cdot 109\) | ||
Dirichlet character group: | $\lbrace$$\chi_{981}(544,·)$, $\chi_{981}(1,·)$, $\chi_{981}(871,·)$, $\chi_{981}(328,·)$, $\chi_{981}(217,·)$, $\chi_{981}(655,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{1089919}a^{5}-\frac{303329}{1089919}a^{4}-\frac{120053}{1089919}a^{3}-\frac{87260}{1089919}a^{2}-\frac{454046}{1089919}a-\frac{390034}{1089919}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{636}{1089919}a^{5}-\frac{1581}{1089919}a^{4}-\frac{59378}{1089919}a^{3}+\frac{88509}{1089919}a^{2}+\frac{1145198}{1089919}a-\frac{1739930}{1089919}$, $\frac{636}{1089919}a^{5}-\frac{1581}{1089919}a^{4}-\frac{59378}{1089919}a^{3}+\frac{88509}{1089919}a^{2}+\frac{1145198}{1089919}a-\frac{650011}{1089919}$, $\frac{17873}{1089919}a^{5}-\frac{142111}{1089919}a^{4}-\frac{746677}{1089919}a^{3}+\frac{6615623}{1089919}a^{2}+\frac{1462635}{1089919}a-\frac{44642437}{1089919}$, $\frac{8620}{1089919}a^{5}+\frac{19701}{1089919}a^{4}-\frac{523729}{1089919}a^{3}-\frac{1227009}{1089919}a^{2}+\frac{7652042}{1089919}a+\frac{19925577}{1089919}$, $\frac{18433}{1089919}a^{5}+\frac{21013}{1089919}a^{4}-\frac{1491298}{1089919}a^{3}-\frac{3012893}{1089919}a^{2}+\frac{28395977}{1089919}a+\frac{74813413}{1089919}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 307.146058197 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 307.146058197 \cdot 3}{2\cdot\sqrt{8496685269}}\cr\approx \mathstrut & 0.319883186347 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{109}) \), \(\Q(\zeta_{9})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{109}) \) $\times$ \(\Q(\zeta_{9})^+\) |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
\(109\) | 109.2.1.1 | $x^{2} + 109$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
109.2.1.1 | $x^{2} + 109$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
109.2.1.1 | $x^{2} + 109$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.109.2t1.a.a | $1$ | $ 109 $ | \(\Q(\sqrt{109}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.981.6t1.b.a | $1$ | $ 3^{2} \cdot 109 $ | 6.6.8496685269.2 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.981.6t1.b.b | $1$ | $ 3^{2} \cdot 109 $ | 6.6.8496685269.2 | $C_6$ (as 6T1) | $0$ | $1$ |