Normalized defining polynomial
\( x^{18} + 3x^{16} - 3x^{14} - 12x^{12} - 12x^{10} + 3x^{8} + 63x^{6} + 3x^{4} + 6x^{2} + 1 \)
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: | \(-485758124386377007104\) \(\medspace = -\,2^{18}\cdot 3^{32}\) | 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: | \(14.10\) | 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: | $2\cdot 3^{16/9}\approx 14.100858664870902$ | ||
Ramified primes: | \(2\), \(3\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{1}{8}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a+\frac{1}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}+\frac{1}{8}a+\frac{1}{4}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{424}a^{16}+\frac{5}{212}a^{14}+\frac{7}{212}a^{12}-\frac{5}{106}a^{10}-\frac{99}{424}a^{8}-\frac{1}{4}a^{7}-\frac{27}{212}a^{6}-\frac{1}{4}a^{5}-\frac{103}{424}a^{4}-\frac{1}{2}a^{3}-\frac{41}{212}a^{2}-\frac{1}{4}a-\frac{197}{424}$, $\frac{1}{424}a^{17}+\frac{5}{212}a^{15}+\frac{7}{212}a^{13}-\frac{5}{106}a^{11}-\frac{99}{424}a^{9}-\frac{1}{4}a^{8}-\frac{27}{212}a^{7}-\frac{1}{4}a^{6}-\frac{103}{424}a^{5}-\frac{1}{2}a^{4}-\frac{41}{212}a^{3}-\frac{1}{4}a^{2}-\frac{197}{424}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{12}{53} a^{17} + \frac{81}{106} a^{15} - \frac{123}{212} a^{13} - \frac{321}{106} a^{11} - \frac{671}{212} a^{9} - \frac{12}{53} a^{7} + \frac{831}{53} a^{5} + \frac{129}{53} a^{3} + \frac{561}{212} a \) (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: | $\frac{11}{212}a^{16}+\frac{1}{53}a^{14}-\frac{29}{106}a^{12}-\frac{2}{53}a^{10}-\frac{29}{212}a^{8}+\frac{127}{106}a^{6}+\frac{139}{212}a^{4}-\frac{239}{106}a^{2}+\frac{59}{212}$, $\frac{9}{106}a^{16}+\frac{21}{212}a^{14}-\frac{33}{106}a^{12}-\frac{95}{212}a^{10}-\frac{48}{53}a^{8}+\frac{75}{53}a^{6}+\frac{93}{53}a^{4}+\frac{273}{212}a^{2}-\frac{12}{53}$, $\frac{195}{424}a^{17}-\frac{23}{424}a^{16}+\frac{143}{106}a^{15}-\frac{71}{424}a^{14}-\frac{331}{212}a^{13}+\frac{51}{212}a^{12}-\frac{1155}{212}a^{11}+\frac{301}{424}a^{10}-\frac{2027}{424}a^{9}+\frac{157}{424}a^{8}+\frac{203}{106}a^{7}-\frac{17}{53}a^{6}+\frac{12563}{424}a^{5}-\frac{1871}{424}a^{4}-\frac{681}{212}a^{3}+\frac{667}{424}a^{2}+\frac{1017}{424}a+\frac{79}{424}$, $\frac{3}{53}a^{17}+\frac{5}{212}a^{16}+\frac{81}{424}a^{15}-\frac{3}{212}a^{14}+\frac{9}{212}a^{13}-\frac{9}{53}a^{12}-\frac{321}{424}a^{11}+\frac{3}{106}a^{10}-\frac{85}{53}a^{9}+\frac{35}{212}a^{8}-\frac{3}{53}a^{7}+\frac{207}{212}a^{6}+\frac{283}{106}a^{5}-\frac{19}{106}a^{4}+\frac{2325}{424}a^{3}-\frac{205}{106}a^{2}-\frac{85}{212}a-\frac{137}{212}$, $\frac{195}{424}a^{17}+\frac{23}{424}a^{16}+\frac{143}{106}a^{15}+\frac{71}{424}a^{14}-\frac{331}{212}a^{13}-\frac{51}{212}a^{12}-\frac{1155}{212}a^{11}-\frac{301}{424}a^{10}-\frac{2027}{424}a^{9}-\frac{157}{424}a^{8}+\frac{203}{106}a^{7}+\frac{17}{53}a^{6}+\frac{12563}{424}a^{5}+\frac{1871}{424}a^{4}-\frac{681}{212}a^{3}-\frac{667}{424}a^{2}+\frac{1017}{424}a-\frac{79}{424}$, $\frac{11}{53}a^{17}+\frac{33}{424}a^{16}+\frac{297}{424}a^{15}+\frac{65}{424}a^{14}-\frac{199}{424}a^{13}-\frac{87}{212}a^{12}-\frac{1177}{424}a^{11}-\frac{289}{424}a^{10}-\frac{1345}{424}a^{9}-\frac{87}{424}a^{8}-\frac{11}{53}a^{7}+\frac{111}{106}a^{6}+\frac{1497}{106}a^{5}+\frac{1689}{424}a^{4}+\frac{1635}{424}a^{3}-\frac{1275}{424}a^{2}+\frac{525}{424}a-\frac{35}{424}$, $\frac{27}{212}a^{17}+\frac{7}{53}a^{16}+\frac{29}{106}a^{15}+\frac{189}{424}a^{14}-\frac{99}{212}a^{13}-\frac{85}{212}a^{12}-\frac{111}{106}a^{11}-\frac{749}{424}a^{10}-\frac{235}{212}a^{9}-\frac{167}{106}a^{8}+\frac{119}{106}a^{7}-\frac{7}{53}a^{6}+\frac{597}{106}a^{5}+\frac{1049}{106}a^{4}-\frac{253}{212}a^{3}-\frac{87}{424}a^{2}+\frac{299}{212}a+\frac{261}{212}$, $\frac{23}{424}a^{17}+\frac{97}{424}a^{16}+\frac{31}{106}a^{15}+\frac{167}{212}a^{14}+\frac{55}{212}a^{13}-\frac{179}{424}a^{12}-\frac{115}{106}a^{11}-\frac{163}{53}a^{10}-\frac{1005}{424}a^{9}-\frac{200}{53}a^{8}-\frac{125}{106}a^{7}-\frac{75}{212}a^{6}+\frac{1553}{424}a^{5}+\frac{6333}{424}a^{4}+\frac{2025}{212}a^{3}+\frac{635}{106}a^{2}-\frac{79}{424}a+\frac{171}{212}$ | 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: | \( 1555.42165689 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 1555.42165689 \cdot 1}{4\cdot\sqrt{485758124386377007104}}\cr\approx \mathstrut & 0.269275554552 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 18T3):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3 \times C_3$ |
Character table for $S_3 \times C_3$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), 3.1.324.1 x3, \(\Q(\zeta_{9})^+\), 6.0.419904.2, 6.0.419904.1, 6.0.419904.3 x2, 9.3.2754990144.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 6 sibling: | 6.0.419904.3 |
Degree 9 sibling: | 9.3.2754990144.1 |
Minimal sibling: | 6.0.419904.3 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.1.0.1}{1} }^{18}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
\(3\) | Deg $18$ | $9$ | $2$ | $32$ |