Normalized defining polynomial
\( x^{6} - 3x^{5} - 21x^{4} + 47x^{3} + 81x^{2} - 105x + 25 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(449163873\) \(\medspace = 3^{7}\cdot 59^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.67\) | 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: | $3^{7/6}59^{1/2}\approx 27.67371536012528$ | ||
Ramified primes: | \(3\), \(59\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{177}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{35}a^{4}-\frac{2}{35}a^{3}+\frac{1}{5}a^{2}-\frac{6}{35}a+\frac{1}{7}$, $\frac{1}{35}a^{5}+\frac{3}{35}a^{3}+\frac{8}{35}a^{2}-\frac{1}{5}a+\frac{2}{7}$
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)
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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: | $\frac{1}{35}a^{4}-\frac{2}{35}a^{3}-\frac{4}{5}a^{2}+\frac{29}{35}a+\frac{29}{7}$, $\frac{1}{35}a^{5}-\frac{3}{35}a^{4}-\frac{26}{35}a^{3}+\frac{57}{35}a^{2}+\frac{151}{35}a-\frac{22}{7}$, $\frac{3}{7}a^{5}-\frac{36}{35}a^{4}-\frac{338}{35}a^{3}+\frac{498}{35}a^{2}+\frac{1546}{35}a-\frac{118}{7}$, $\frac{2}{35}a^{5}-\frac{2}{35}a^{4}-\frac{12}{7}a^{3}+\frac{37}{35}a^{2}+\frac{418}{35}a-\frac{47}{7}$, $\frac{21}{5}a^{5}-\frac{393}{35}a^{4}-\frac{3218}{35}a^{3}+168a^{2}+\frac{13999}{35}a-\frac{2143}{7}$ | 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: | \( 470.319992162 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 470.319992162 \cdot 1}{2\cdot\sqrt{449163873}}\cr\approx \mathstrut & 0.710135149276 \end{aligned}\]
Galois group
A solvable group of order 6 |
The 3 conjugacy class representatives for $S_3$ |
Character table for $S_3$ |
Intermediate fields
\(\Q(\sqrt{177}) \), 3.3.1593.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 3.3.1593.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\) |
Degree 3 sibling: | 3.3.1593.1 |
Minimal sibling: | 3.3.1593.1 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.1.0.1}{1} }^{6}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | R |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.7.4 | $x^{6} + 3 x^{2} + 3$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ |
\(59\) | 59.2.1.1 | $x^{2} + 118$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
59.2.1.1 | $x^{2} + 118$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.2.1.1 | $x^{2} + 118$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |