Normalized defining polynomial
\( x^{6} - 2x^{5} + 5x^{4} - 60x^{3} + 60x^{2} - 112x + 79 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(8760065625\) \(\medspace = 3^{3}\cdot 5^{5}\cdot 47^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.40\) | 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: | $3^{1/2}5^{5/6}47^{1/2}\approx 45.40300106204512$ | ||
Ramified primes: | \(3\), \(5\), \(47\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{705}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{694}a^{5}-\frac{14}{347}a^{4}+\frac{39}{694}a^{3}-\frac{33}{694}a^{2}-\frac{123}{694}a+\frac{155}{347}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | 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: | $\frac{1}{694}a^{5}-\frac{14}{347}a^{4}+\frac{39}{694}a^{3}-\frac{33}{694}a^{2}+\frac{1265}{694}a-\frac{539}{347}$, $\frac{247}{347}a^{5}-\frac{323}{347}a^{4}+\frac{2263}{694}a^{3}-\frac{28447}{694}a^{2}+\frac{3972}{347}a-\frac{49855}{694}$, $\frac{13}{347}a^{5}-\frac{1075}{694}a^{4}+\frac{4831}{694}a^{3}-\frac{2858}{347}a^{2}+\frac{8947}{694}a-\frac{2563}{347}$ | 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: | \( 290.982584503 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 290.982584503 \cdot 2}{2\cdot\sqrt{8760065625}}\cr\approx \mathstrut & 0.490945230427 \end{aligned}\]
Galois group
A solvable group of order 12 |
The 6 conjugacy class representatives for $D_{6}$ |
Character table for $D_{6}$ |
Intermediate fields
\(\Q(\sqrt{705}) \), 3.1.1175.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Galois closure: | deg 12 |
Twin sextic algebra: | 3.1.1175.1 $\times$ \(\Q(\sqrt{-15}) \) $\times$ \(\Q\) |
Degree 6 sibling: | 6.0.186384375.1 |
Minimal sibling: | 6.0.186384375.1 |
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 | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.3.1 | $x^{6} + 18 x^{2} - 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(5\) | 5.6.5.1 | $x^{6} + 5$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
\(47\) | 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.47.2t1.a.a | $1$ | $ 47 $ | \(\Q(\sqrt{-47}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.705.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 47 $ | \(\Q(\sqrt{705}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.1175.3t2.a.a | $2$ | $ 5^{2} \cdot 47 $ | 3.1.1175.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.10575.6t3.b.a | $2$ | $ 3^{2} \cdot 5^{2} \cdot 47 $ | 6.2.8760065625.1 | $D_{6}$ (as 6T3) | $1$ | $0$ |