Normalized defining polynomial
\( x^{6} - x^{5} + 31x^{4} + 84x^{3} + 873x^{2} + 810x + 729 \)
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: | \(-1851523947\) \(\medspace = -\,3^{3}\cdot 7^{4}\cdot 13^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.04\) | 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: | $3^{1/2}7^{2/3}13^{2/3}\approx 35.04194650073235$ | ||
Ramified primes: | \(3\), \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(273=3\cdot 7\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{273}(1,·)$, $\chi_{273}(100,·)$, $\chi_{273}(263,·)$, $\chi_{273}(92,·)$, $\chi_{273}(172,·)$, $\chi_{273}(191,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-3}) \), 6.0.1851523947.1$^{3}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{9}a^{4}-\frac{1}{9}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{250857}a^{5}+\frac{9260}{250857}a^{4}-\frac{36203}{250857}a^{3}+\frac{39970}{83619}a^{2}-\frac{6284}{27873}a+\frac{310}{3097}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{6}\times C_{6}$, which has order $36$
Unit group
Rank: | $2$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -\frac{310}{250857} a^{5} + \frac{319}{250857} a^{4} - \frac{9889}{250857} a^{3} - \frac{5797}{83619} a^{2} - \frac{30943}{27873} a - \frac{93}{3097} \) (order $6$) | 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: | $\frac{1640}{250857}a^{5}-\frac{4385}{250857}a^{4}+\frac{52316}{250857}a^{3}+\frac{30668}{83619}a^{2}+\frac{109451}{27873}a+\frac{492}{3097}$, $\frac{2570}{250857}a^{5}-\frac{5342}{250857}a^{4}+\frac{81983}{250857}a^{3}+\frac{48059}{83619}a^{2}+\frac{230153}{27873}a+\frac{771}{3097}$ | 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: | \( 31.7983118742 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 31.7983118742 \cdot 36}{6\cdot\sqrt{1851523947}}\cr\approx \mathstrut & 1.09984188117 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 3.3.8281.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 3.3.8281.2 $\times$ \(\Q(\sqrt{-3}) \) $\times$ \(\Q\) |
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 | ${\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(7\) | 7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(13\) | 13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
13.3.2.2 | $x^{3} + 13$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.91.3t1.a.a | $1$ | $ 7 \cdot 13 $ | 3.3.8281.2 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.273.6t1.d.a | $1$ | $ 3 \cdot 7 \cdot 13 $ | 6.0.1851523947.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
* | 1.91.3t1.a.b | $1$ | $ 7 \cdot 13 $ | 3.3.8281.2 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.273.6t1.d.b | $1$ | $ 3 \cdot 7 \cdot 13 $ | 6.0.1851523947.1 | $C_6$ (as 6T1) | $0$ | $-1$ |