Normalized defining polynomial
\( x^{10} - 2x^{9} + 3x^{8} - 12x^{7} - 4x^{6} - 6x^{5} - 37x^{4} - 24x^{3} - 24x^{2} - 64x - 16 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(742299263709184\) \(\medspace = 2^{12}\cdot 5659^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(30.69\) | 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))
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Galois root discriminant: | $2^{7/4}5659^{1/2}\approx 253.03018870434167$ | ||
Ramified primes: | \(2\), \(5659\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{5659}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{38}a^{7}-\frac{1}{19}a^{6}-\frac{5}{38}a^{5}+\frac{1}{19}a^{4}+\frac{1}{19}a^{3}-\frac{6}{19}a^{2}-\frac{1}{2}a-\frac{4}{19}$, $\frac{1}{152}a^{8}-\frac{1}{76}a^{7}+\frac{71}{152}a^{6}-\frac{9}{38}a^{5}+\frac{5}{19}a^{4}+\frac{13}{76}a^{3}+\frac{1}{8}a^{2}-\frac{1}{19}a-\frac{1}{2}$, $\frac{1}{608}a^{9}-\frac{5}{608}a^{7}+\frac{125}{304}a^{6}+\frac{3}{76}a^{5}+\frac{7}{16}a^{4}+\frac{79}{608}a^{3}-\frac{85}{304}a^{2}+\frac{15}{152}a+\frac{15}{76}$
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: | $\frac{7}{304}a^{9}-\frac{1}{38}a^{8}+\frac{21}{304}a^{7}-\frac{3}{8}a^{6}-\frac{3}{19}a^{5}-\frac{101}{152}a^{4}-\frac{183}{304}a^{3}+\frac{1}{152}a^{2}-\frac{221}{76}a-\frac{11}{38}$, $\frac{4}{19}a^{9}-\frac{47}{76}a^{8}+a^{7}-\frac{221}{76}a^{6}+\frac{47}{38}a^{5}-\frac{2}{19}a^{4}-\frac{297}{38}a^{3}+\frac{51}{76}a^{2}+\frac{41}{38}a-\frac{233}{19}$, $\frac{7}{304}a^{9}-\frac{1}{38}a^{8}+\frac{21}{304}a^{7}-\frac{3}{8}a^{6}-\frac{3}{19}a^{5}-\frac{101}{152}a^{4}-\frac{183}{304}a^{3}+\frac{1}{152}a^{2}-\frac{69}{76}a-\frac{11}{38}$, $\frac{3}{152}a^{9}-\frac{3}{19}a^{8}+\frac{37}{152}a^{7}-\frac{25}{76}a^{6}+\frac{39}{38}a^{5}+\frac{75}{76}a^{4}-\frac{227}{152}a^{3}+\frac{177}{76}a^{2}+\frac{75}{19}a+\frac{3}{19}$, $\frac{1}{38}a^{7}-\frac{1}{19}a^{6}-\frac{5}{38}a^{5}+\frac{1}{19}a^{4}+\frac{1}{19}a^{3}-\frac{6}{19}a^{2}-\frac{1}{2}a-\frac{4}{19}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 7444.62604551 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 7444.62604551 \cdot 1}{2\cdot\sqrt{742299263709184}}\cr\approx \mathstrut & 0.851731759524 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_{6}$ |
Character table for $S_{6}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 6 siblings: | 6.2.362176.2, 6.2.46393703981824.2 |
Degree 12 siblings: | data not computed |
Degree 15 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.2.362176.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.7 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $[2, 2]^{3}$ |
2.6.6.6 | $x^{6} - 4 x^{5} + 30 x^{4} - 16 x^{3} + 164 x^{2} + 160 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ | |
\(5659\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |