Normalized defining polynomial
\( x^{6} - x^{5} + 2x^{4} + 8x^{3} - x^{2} - 5x + 7 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-2476099\) \(\medspace = -\,19^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.63\) | 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: | $19^{5/6}\approx 11.631293127556278$ | ||
Ramified primes: | \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{19}(1,·)$, $\chi_{19}(18,·)$, $\chi_{19}(7,·)$, $\chi_{19}(8,·)$, $\chi_{19}(11,·)$, $\chi_{19}(12,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-19}) \), 6.0.2476099.1$^{3}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{77}a^{5}+\frac{24}{77}a^{4}-\frac{2}{11}a^{3}-\frac{34}{77}a^{2}-\frac{4}{77}a-\frac{4}{11}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $2$ | 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: | $\frac{15}{77}a^{5}-\frac{25}{77}a^{4}+\frac{3}{11}a^{3}+\frac{106}{77}a^{2}-\frac{137}{77}a-\frac{27}{11}$, $\frac{9}{77}a^{5}-\frac{15}{77}a^{4}+\frac{4}{11}a^{3}+\frac{79}{77}a^{2}-\frac{113}{77}a+\frac{8}{11}$ | 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: | \( 7.80862678603 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 7.80862678603 \cdot 1}{2\cdot\sqrt{2476099}}\cr\approx \mathstrut & 0.615460624554 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{-19}) \), 3.3.361.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 3.3.361.1 $\times$ \(\Q(\sqrt{-19}) \) $\times$ \(\Q\) |
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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.1.0.1}{1} }^{6}$ | ${\href{/padicField/11.1.0.1}{1} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(19\) | 19.6.5.5 | $x^{6} + 19$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.19.6t1.a.a | $1$ | $ 19 $ | 6.0.2476099.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
* | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.19.6t1.a.b | $1$ | $ 19 $ | 6.0.2476099.1 | $C_6$ (as 6T1) | $0$ | $-1$ |