Normalized defining polynomial
\( x^{6} - x^{5} - 27x^{4} + 27x^{3} + 197x^{2} - 197x - 251 \)
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: | \(56723625\) \(\medspace = 3^{3}\cdot 5^{3}\cdot 7^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.60\) | 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: | $3^{1/2}5^{1/2}7^{5/6}\approx 19.601711648537535$ | ||
Ramified primes: | \(3\), \(5\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{105}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(105=3\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{105}(16,·)$, $\chi_{105}(1,·)$, $\chi_{105}(104,·)$, $\chi_{105}(89,·)$, $\chi_{105}(59,·)$, $\chi_{105}(46,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{13}a^{4}+\frac{2}{13}a^{3}-\frac{3}{13}a^{2}+\frac{2}{13}a+\frac{6}{13}$, $\frac{1}{13}a^{5}+\frac{6}{13}a^{3}-\frac{5}{13}a^{2}+\frac{2}{13}a+\frac{1}{13}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{1}{13}a^{5}+\frac{3}{13}a^{4}-\frac{27}{13}a^{3}-\frac{53}{13}a^{2}+\frac{164}{13}a+\frac{136}{13}$, $\frac{3}{13}a^{4}-\frac{7}{13}a^{3}-\frac{48}{13}a^{2}+\frac{84}{13}a+\frac{96}{13}$, $\frac{1}{13}a^{5}+\frac{9}{13}a^{4}-\frac{41}{13}a^{3}-\frac{149}{13}a^{2}+\frac{345}{13}a+\frac{302}{13}$, $\frac{2}{13}a^{5}-\frac{7}{13}a^{4}-\frac{28}{13}a^{3}+\frac{102}{13}a^{2}+\frac{16}{13}a-\frac{157}{13}$, $\frac{1}{13}a^{5}+\frac{1}{13}a^{4}-\frac{18}{13}a^{3}-\frac{21}{13}a^{2}+\frac{69}{13}a+\frac{111}{13}$ | 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: | \( 43.080677479 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 43.080677479 \cdot 2}{2\cdot\sqrt{56723625}}\cr\approx \mathstrut & 0.36608379903 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{105}) \), \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q(\zeta_{7})^+\) $\times$ \(\Q(\sqrt{105}) \) $\times$ \(\Q\) |
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 | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | R | R | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(5\) | 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(7\) | 7.6.5.2 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.105.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 7 $ | \(\Q(\sqrt{105}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.105.6t1.b.a | $1$ | $ 3 \cdot 5 \cdot 7 $ | 6.6.56723625.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.105.6t1.b.b | $1$ | $ 3 \cdot 5 \cdot 7 $ | 6.6.56723625.1 | $C_6$ (as 6T1) | $0$ | $1$ |