Normalized defining polynomial
\( x^{6} - x^{5} - 215x^{4} + 143x^{3} + 14775x^{2} - 5041x - 323611 \)
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: | \(53273486441\) \(\medspace = 7^{4}\cdot 281^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(61.34\) | 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: | $7^{2/3}281^{1/2}\approx 61.34114146731612$ | ||
Ramified primes: | \(7\), \(281\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{281}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1967=7\cdot 281\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1967}(1,·)$, $\chi_{1967}(561,·)$, $\chi_{1967}(1404,·)$, $\chi_{1967}(842,·)$, $\chi_{1967}(844,·)$, $\chi_{1967}(282,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{21096307}a^{5}+\frac{4504414}{21096307}a^{4}-\frac{3003181}{21096307}a^{3}-\frac{9702669}{21096307}a^{2}-\frac{8290486}{21096307}a+\frac{7105775}{21096307}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{562}{21096307}a^{5}-\frac{76172}{21096307}a^{4}-\frac{83162}{21096307}a^{3}+\frac{11043535}{21096307}a^{2}+\frac{3030715}{21096307}a-\frac{373489999}{21096307}$, $\frac{1110}{21096307}a^{5}+\frac{74781}{21096307}a^{4}-\frac{314404}{21096307}a^{3}-\frac{10846020}{21096307}a^{2}+\frac{16646699}{21096307}a+\frac{356028651}{21096307}$, $\frac{173761}{21096307}a^{5}-\frac{2604953}{21096307}a^{4}-\frac{18580096}{21096307}a^{3}+\frac{303446708}{21096307}a^{2}+\frac{462304403}{21096307}a-\frac{8435512814}{21096307}$, $\frac{535086}{21096307}a^{5}-\frac{4205146}{21096307}a^{4}-\frac{75500683}{21096307}a^{3}+\frac{566215841}{21096307}a^{2}+\frac{2350134241}{21096307}a-\frac{16357423192}{21096307}$, $\frac{212160080}{21096307}a^{5}-\frac{176503990}{21096307}a^{4}-\frac{50447149740}{21096307}a^{3}+\frac{25062678350}{21096307}a^{2}+\frac{5173777457690}{21096307}a-\frac{23313812826429}{21096307}$ | 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: | \( 2176.9888573 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 2176.9888573 \cdot 1}{2\cdot\sqrt{53273486441}}\cr\approx \mathstrut & 0.30182181374 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{281}) \), \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{281}) \) $\times$ \(\Q(\zeta_{7})^+\) |
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 | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
\(281\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.281.2t1.a.a | $1$ | $ 281 $ | \(\Q(\sqrt{281}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.1967.6t1.b.a | $1$ | $ 7 \cdot 281 $ | 6.6.53273486441.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.1967.6t1.b.b | $1$ | $ 7 \cdot 281 $ | 6.6.53273486441.1 | $C_6$ (as 6T1) | $0$ | $1$ |