Normalized defining polynomial
\( x^{6} - x^{5} - 95x^{4} + 530x^{3} - 925x^{2} + 367x + 187 \)
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: | \(629763392149\) \(\medspace = 229^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(92.58\) | 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: | $229^{5/6}\approx 92.58264171776156$ | ||
Ramified primes: | \(229\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{229}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(229\) | ||
Dirichlet character group: | $\lbrace$$\chi_{229}(1,·)$, $\chi_{229}(228,·)$, $\chi_{229}(134,·)$, $\chi_{229}(135,·)$, $\chi_{229}(94,·)$, $\chi_{229}(95,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{24}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{7}{24}a+\frac{1}{24}$, $\frac{1}{144}a^{5}-\frac{1}{72}a^{4}-\frac{7}{48}a^{3}+\frac{47}{144}a^{2}+\frac{1}{4}a-\frac{29}{144}$
Monogenic: | No | |
Index: | $4$ | |
Inessential primes: | $2$ |
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: | $\frac{5}{72}a^{5}+\frac{11}{72}a^{4}-\frac{73}{12}a^{3}+\frac{1261}{72}a^{2}-\frac{229}{24}a-\frac{38}{9}$, $\frac{1}{144}a^{5}-\frac{1}{72}a^{4}-\frac{31}{48}a^{3}+\frac{623}{144}a^{2}-\frac{39}{4}a+\frac{1051}{144}$, $\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{23}{2}a^{3}+\frac{349}{8}a^{2}-\frac{305}{8}a+\frac{3}{4}$, $\frac{55}{144}a^{5}+\frac{4}{9}a^{4}-\frac{1627}{48}a^{3}+\frac{18965}{144}a^{2}-\frac{4673}{24}a+\frac{17659}{144}$, $\frac{125}{144}a^{5}+\frac{13}{72}a^{4}-\frac{3935}{48}a^{3}+\frac{52315}{144}a^{2}-\frac{2245}{6}a-\frac{19405}{144}$ | 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: | \( 8119.61036943 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 8119.61036943 \cdot 3}{2\cdot\sqrt{629763392149}}\cr\approx \mathstrut & 0.982240227712 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{229}) \), 3.3.52441.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 3.3.52441.1 $\times$ \(\Q(\sqrt{229}) \) $\times$ \(\Q\) |
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 | ${\href{/padicField/2.2.0.1}{2} }^{3}$ | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.1.0.1}{1} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.1.0.1}{1} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(229\) | Deg $6$ | $6$ | $1$ | $5$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.229.2t1.a.a | $1$ | $ 229 $ | \(\Q(\sqrt{229}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.229.3t1.a.a | $1$ | $ 229 $ | 3.3.52441.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.229.6t1.a.a | $1$ | $ 229 $ | 6.6.629763392149.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.229.3t1.a.b | $1$ | $ 229 $ | 3.3.52441.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.229.6t1.a.b | $1$ | $ 229 $ | 6.6.629763392149.1 | $C_6$ (as 6T1) | $0$ | $1$ |