Normalized defining polynomial
\( x^{18} + 108x^{12} - 424x^{9} + 576x^{6} - 288x^{3} + 64 \)
Invariants
| Degree: | $18$ |
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| Signature: | $(0, 9)$ |
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| Discriminant: |
\(-12100864846032214829641728\)
\(\medspace = -\,2^{12}\cdot 3^{45}\)
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| Root discriminant: | \(24.75\) |
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| Galois root discriminant: | $2^{2/3}3^{49/18}\approx 31.5876084551639$ | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_9$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\zeta_{9})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{8}a^{9}$, $\frac{1}{8}a^{10}$, $\frac{1}{8}a^{11}$, $\frac{1}{16}a^{12}$, $\frac{1}{16}a^{13}$, $\frac{1}{16}a^{14}$, $\frac{1}{66272}a^{15}-\frac{30}{2071}a^{12}+\frac{273}{8284}a^{9}+\frac{221}{2071}a^{6}+\frac{271}{4142}a^{3}+\frac{384}{2071}$, $\frac{1}{66272}a^{16}-\frac{30}{2071}a^{13}+\frac{273}{8284}a^{10}+\frac{221}{2071}a^{7}+\frac{271}{4142}a^{4}+\frac{384}{2071}a$, $\frac{1}{66272}a^{17}-\frac{30}{2071}a^{14}+\frac{273}{8284}a^{11}+\frac{221}{2071}a^{8}+\frac{271}{4142}a^{5}+\frac{384}{2071}a^{2}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( \frac{571}{33136} a^{15} + \frac{655}{33136} a^{12} + \frac{15613}{8284} a^{9} - \frac{10635}{2071} a^{6} + \frac{17471}{4142} a^{3} - \frac{524}{2071} \)
(order $18$)
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| Fundamental units: |
$\frac{483}{33136}a^{15}+\frac{14}{2071}a^{12}+\frac{13127}{8284}a^{9}-\frac{22435}{4142}a^{6}+\frac{27763}{4142}a^{3}-\frac{1836}{2071}$, $\frac{2195}{66272}a^{15}+\frac{539}{33136}a^{12}+\frac{14855}{4142}a^{9}-\frac{50813}{4142}a^{6}+\frac{27157}{2071}a^{3}-\frac{6230}{2071}$, $\frac{1227}{66272}a^{17}-\frac{211}{16568}a^{16}-\frac{881}{33136}a^{15}-\frac{199}{8284}a^{14}-\frac{199}{8284}a^{13}-\frac{1279}{33136}a^{12}+\frac{16037}{8284}a^{11}-\frac{23843}{16568}a^{10}-\frac{48741}{16568}a^{9}-\frac{43759}{4142}a^{8}+\frac{22245}{8284}a^{7}+\frac{14443}{2071}a^{6}+\frac{28537}{2071}a^{5}-\frac{8741}{2071}a^{4}-\frac{28095}{4142}a^{3}-\frac{11375}{2071}a^{2}+\frac{7264}{2071}a+\frac{2680}{2071}$, $\frac{7825}{66272}a^{17}+\frac{75}{16568}a^{16}-\frac{111}{1744}a^{15}+\frac{397}{16568}a^{14}+\frac{485}{16568}a^{13}-\frac{21}{872}a^{12}+\frac{211219}{16568}a^{11}+\frac{8475}{16568}a^{10}-\frac{5999}{872}a^{9}-\frac{196665}{4142}a^{8}+\frac{10467}{8284}a^{7}+\frac{5317}{218}a^{6}+\frac{119018}{2071}a^{5}-\frac{15267}{2071}a^{4}-\frac{2940}{109}a^{3}-\frac{39570}{2071}a^{2}+\frac{15792}{2071}a+\frac{753}{109}$, $\frac{1453}{66272}a^{17}-\frac{25}{4142}a^{16}+\frac{655}{66272}a^{15}+\frac{487}{33136}a^{14}-\frac{603}{33136}a^{13}+\frac{49}{4142}a^{12}+\frac{19747}{8284}a^{11}-\frac{2825}{4142}a^{10}+\frac{8993}{8284}a^{9}-\frac{63769}{8284}a^{8}+\frac{4683}{8284}a^{7}-\frac{23641}{8284}a^{6}+\frac{33409}{4142}a^{5}+\frac{1717}{2071}a^{4}+\frac{11825}{4142}a^{3}-\frac{5360}{2071}a^{2}-\frac{346}{2071}a-\frac{1142}{2071}$, $\frac{7}{2071}a^{17}+\frac{25}{4142}a^{16}+\frac{659}{66272}a^{15}+\frac{43}{8284}a^{14}+\frac{603}{33136}a^{13}+\frac{543}{33136}a^{12}+\frac{791}{2071}a^{11}+\frac{2825}{4142}a^{10}+\frac{18099}{16568}a^{9}-\frac{7013}{8284}a^{8}-\frac{4683}{8284}a^{7}-\frac{20105}{8284}a^{6}+\frac{3429}{2071}a^{5}-\frac{1717}{2071}a^{4}+\frac{4625}{4142}a^{3}-\frac{3037}{2071}a^{2}+\frac{346}{2071}a+\frac{394}{2071}$, $\frac{3241}{66272}a^{17}+\frac{2323}{66272}a^{16}+\frac{2941}{66272}a^{15}+\frac{107}{2071}a^{14}+\frac{1229}{33136}a^{13}+\frac{371}{16568}a^{12}+\frac{43967}{8284}a^{11}+\frac{15759}{4142}a^{10}+\frac{79459}{16568}a^{9}-\frac{31370}{2071}a^{8}-\frac{89953}{8284}a^{7}-\frac{135947}{8284}a^{6}+\frac{37485}{4142}a^{5}+\frac{13436}{2071}a^{4}+\frac{68019}{4142}a^{3}+\frac{12299}{2071}a^{2}+\frac{7715}{2071}a-\frac{3493}{2071}$, $\frac{185}{16568}a^{17}+\frac{211}{16568}a^{16}+\frac{439}{66272}a^{15}-\frac{1059}{33136}a^{14}+\frac{199}{8284}a^{13}+\frac{261}{16568}a^{12}+\frac{9417}{8284}a^{11}+\frac{23843}{16568}a^{10}+\frac{2971}{4142}a^{9}-\frac{68619}{8284}a^{8}-\frac{22245}{8284}a^{7}-\frac{2389}{2071}a^{6}+\frac{25714}{2071}a^{5}+\frac{8741}{2071}a^{4}-\frac{5752}{2071}a^{3}-\frac{5780}{2071}a^{2}-\frac{7264}{2071}a+\frac{2896}{2071}$
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| Regulator: | \( 957708.9948188227 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 957708.9948188227 \cdot 1}{18\cdot\sqrt{12100864846032214829641728}}\cr\approx \mathstrut & 0.233438063196957 \end{aligned}\]
Galois group
$S_3\times C_9$ (as 18T16):
| A solvable group of order 54 |
| The 27 conjugacy class representatives for $C_9\times S_3$ |
| Character table for $C_9\times S_3$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 27 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
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 | $18$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | $18$ | $18$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{9}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.3.12a1.1 | $x^{18} + 3 x^{16} + 3 x^{15} + 3 x^{14} + 9 x^{13} + 7 x^{12} + 9 x^{11} + 15 x^{10} + 10 x^{9} + 12 x^{8} + 15 x^{7} + 9 x^{6} + 9 x^{5} + 9 x^{4} + 4 x^{3} + 3 x^{2} + 5 x + 1$ | $3$ | $6$ | $12$ | $C_9\times S_3$ | $$[\ ]_{3}^{18}$$ |
|
\(3\)
| 3.1.18.45a2.41 | $x^{18} + 9 x^{17} + 18 x^{16} + 9 x^{14} + 9 x^{13} + 6 x^{12} + 18 x^{10} + 3$ | $18$ | $1$ | $45$ | $C_9\times S_3$ | not computed |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *54 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *54 | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *54 | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *54 | 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| *54 | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| *54 | 1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| 1.27.18t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.9t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.18t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.18t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.9t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.18t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.18t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.18t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ | |
| 1.27.9t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.9t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.9t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 1.27.9t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ | |
| 2.972.3t2.c.a | $2$ | $ 2^{2} \cdot 3^{5}$ | \(\Q(\sqrt[3]{12})\) | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.972.6t5.c.a | $2$ | $ 2^{2} \cdot 3^{5}$ | 6.0.2834352.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
| 2.972.6t5.c.b | $2$ | $ 2^{2} \cdot 3^{5}$ | 6.0.2834352.3 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
| *54 | 2.2916.18t16.f.a | $2$ | $ 2^{2} \cdot 3^{6}$ | 18.0.12100864846032214829641728.3 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
| *54 | 2.2916.18t16.f.b | $2$ | $ 2^{2} \cdot 3^{6}$ | 18.0.12100864846032214829641728.3 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
| *54 | 2.2916.18t16.f.c | $2$ | $ 2^{2} \cdot 3^{6}$ | 18.0.12100864846032214829641728.3 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
| *54 | 2.2916.18t16.f.d | $2$ | $ 2^{2} \cdot 3^{6}$ | 18.0.12100864846032214829641728.3 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
| *54 | 2.2916.18t16.f.e | $2$ | $ 2^{2} \cdot 3^{6}$ | 18.0.12100864846032214829641728.3 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
| *54 | 2.2916.18t16.f.f | $2$ | $ 2^{2} \cdot 3^{6}$ | 18.0.12100864846032214829641728.3 | $C_9\times S_3$ (as 18T16) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.