Normalized defining polynomial
\( x^{18} - 6x^{12} - 6x^{9} + 13x^{6} + 26x^{3} + 13 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-82845576677855741607936\) \(\medspace = -\,2^{18}\cdot 3^{18}\cdot 13^{8}\) | 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: | \(18.76\) | 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: | not computed | ||
Ramified primes: | \(2\), \(3\), \(13\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}$, $\frac{1}{34}a^{15}+\frac{7}{34}a^{12}-\frac{4}{17}a^{9}+\frac{3}{17}a^{6}+\frac{2}{17}a^{3}+\frac{3}{34}$, $\frac{1}{34}a^{16}+\frac{7}{34}a^{13}-\frac{4}{17}a^{10}+\frac{3}{17}a^{7}+\frac{2}{17}a^{4}+\frac{3}{34}a$, $\frac{1}{34}a^{17}+\frac{7}{34}a^{14}-\frac{4}{17}a^{11}+\frac{3}{17}a^{8}+\frac{2}{17}a^{5}+\frac{3}{34}a^{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{5}{2} a^{9} + \frac{13}{2} a^{3} + 5 \) (order $4$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{2}{17}a^{15}-\frac{3}{17}a^{12}-\frac{15}{34}a^{9}-\frac{5}{17}a^{6}+\frac{67}{34}a^{3}+\frac{63}{34}$, $\frac{3}{34}a^{15}+\frac{2}{17}a^{12}-\frac{12}{17}a^{9}-\frac{33}{34}a^{6}+\frac{63}{34}a^{3}+\frac{111}{34}$, $\frac{6}{17}a^{16}-\frac{9}{17}a^{13}-\frac{45}{34}a^{10}+\frac{2}{17}a^{7}+\frac{133}{34}a^{4}+\frac{87}{34}a+1$, $\frac{9}{34}a^{16}+\frac{9}{34}a^{15}-\frac{5}{34}a^{13}-\frac{5}{34}a^{12}-\frac{55}{34}a^{10}-\frac{55}{34}a^{9}-\frac{7}{17}a^{7}-\frac{7}{17}a^{6}+\frac{121}{34}a^{4}+\frac{121}{34}a^{3}+\frac{73}{17}a+\frac{73}{17}$, $\frac{21}{34}a^{16}-\frac{1}{34}a^{15}-\frac{23}{34}a^{13}+\frac{5}{17}a^{12}-\frac{50}{17}a^{10}-\frac{9}{34}a^{9}-\frac{5}{17}a^{7}-\frac{23}{34}a^{6}+\frac{144}{17}a^{4}-\frac{2}{17}a^{3}+\frac{199}{34}a+\frac{41}{17}$, $\frac{1}{34}a^{17}+\frac{2}{17}a^{16}+\frac{6}{17}a^{15}+\frac{7}{34}a^{14}-\frac{3}{17}a^{13}-\frac{9}{17}a^{12}-\frac{25}{34}a^{11}-\frac{15}{34}a^{10}-\frac{45}{34}a^{9}-\frac{14}{17}a^{8}-\frac{5}{17}a^{7}+\frac{2}{17}a^{6}+\frac{55}{34}a^{5}+\frac{67}{34}a^{4}+\frac{133}{34}a^{3}+\frac{44}{17}a^{2}+\frac{63}{34}a+\frac{87}{34}$, $\frac{1}{34}a^{16}+\frac{4}{17}a^{15}+\frac{7}{34}a^{13}-\frac{6}{17}a^{12}-\frac{25}{34}a^{10}-\frac{15}{17}a^{9}-\frac{14}{17}a^{7}+\frac{7}{17}a^{6}+\frac{55}{34}a^{4}+\frac{33}{17}a^{3}+a^{2}+\frac{44}{17}a+\frac{12}{17}$, $\frac{5}{17}a^{17}-\frac{3}{17}a^{15}-\frac{15}{34}a^{14}+\frac{9}{34}a^{12}-\frac{23}{17}a^{11}+\frac{31}{34}a^{9}+\frac{9}{34}a^{8}-\frac{19}{34}a^{6}+\frac{125}{34}a^{5}-\frac{46}{17}a^{3}+\frac{49}{17}a^{2}+a-\frac{35}{34}$ (assuming GRH) | 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: | \( 20735.3162984 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 20735.3162984 \cdot 1}{4\cdot\sqrt{82845576677855741607936}}\cr\approx \mathstrut & 0.274874736302 \end{aligned}\] (assuming GRH)
Galois group
$\He_3^2:S_3^2$ (as 18T650):
A solvable group of order 26244 |
The 109 conjugacy class representatives for $\He_3^2:S_3^2$ are not computed |
Character table for $\He_3^2:S_3^2$ is not computed |
Intermediate fields
\(\Q(\sqrt{-1}) \), 6.0.10816.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 18 siblings: | data not computed |
Degree 36 siblings: | data not computed |
Minimal sibling: | 18.0.82845576677855741607936.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | $18$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | $18$ | $18$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | $18$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | $18$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
\(3\) | Deg $18$ | $3$ | $6$ | $18$ | |||
\(13\) | 13.9.8.1 | $x^{9} + 26$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |