Normalized defining polynomial
\( x^{18} - 6x^{16} + 5x^{14} + 35x^{12} - 88x^{10} + 52x^{8} + 36x^{6} - 46x^{4} + 13x^{2} - 1 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[14, 2]$ |
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| Discriminant: |
\(70853239618582582520971264\)
\(\medspace = 2^{18}\cdot 37^{4}\cdot 229^{6}\)
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| Root discriminant: | \(27.30\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(37\), \(229\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
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Unit group
| Rank: | $15$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$a^{17}-6a^{15}+5a^{13}+35a^{11}-88a^{9}+52a^{7}+36a^{5}-46a^{3}+13a$, $a^{2}-1$, $a^{17}-7a^{15}+9a^{13}+39a^{11}-119a^{9}+75a^{7}+59a^{5}-65a^{3}+12a$, $a^{15}-3a^{13}-6a^{11}+24a^{9}-7a^{7}-22a^{5}+6a^{3}+4a$, $a^{17}-7a^{15}+9a^{13}+39a^{11}-119a^{9}+75a^{7}+59a^{5}-65a^{3}+11a$, $a^{15}-4a^{13}-3a^{11}+29a^{9}-30a^{7}-8a^{5}+20a^{3}-6a$, $a^{12}-2a^{10}-7a^{8}+16a^{6}+a^{4}-13a^{2}+3$, $a^{16}-4a^{14}-4a^{12}+31a^{10}-23a^{8}-23a^{6}+19a^{4}+a^{2}$, $a-1$, $a^{17}+3a^{16}-5a^{15}-14a^{14}-2a^{12}+35a^{11}+97a^{10}-53a^{9}-143a^{8}-a^{7}+6a^{6}+35a^{5}+92a^{4}-11a^{3}-40a^{2}+2a+3$, $a^{2}+a-1$, $3a^{15}+2a^{14}-11a^{13}-9a^{12}-12a^{11}-5a^{10}+83a^{9}+67a^{8}-67a^{7}-68a^{6}-45a^{5}-32a^{4}+48a^{3}+46a^{2}-4a-6$, $a^{17}-6a^{15}-2a^{14}+5a^{13}+8a^{12}+35a^{11}+6a^{10}-88a^{9}-59a^{8}+52a^{7}+60a^{6}+36a^{5}+23a^{4}-46a^{3}-42a^{2}+13a+8$, $3a^{17}-18a^{15}+2a^{14}+13a^{13}-7a^{12}+112a^{11}-9a^{10}-255a^{9}+53a^{8}+103a^{7}-37a^{6}+145a^{5}-30a^{4}-108a^{3}+26a^{2}+13a-2$, $a^{17}+a^{16}-6a^{15}-5a^{14}+6a^{13}+a^{12}+33a^{11}+32a^{10}-95a^{9}-59a^{8}+68a^{7}+22a^{6}+37a^{5}+28a^{4}-59a^{3}-26a^{2}+15a+6$
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| Regulator: | \( 4596460.1775 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 4596460.1775 \cdot 1}{2\cdot\sqrt{70853239618582582520971264}}\cr\approx \mathstrut & 0.17660106817 \end{aligned}\] (assuming GRH)
Galois group
$A_4^3.S_4$ (as 18T711):
| A solvable group of order 41472 |
| The 96 conjugacy class representatives for $A_4^3.S_4$ |
| Character table for $A_4^3.S_4$ |
Intermediate fields
| 3.3.229.1, 9.9.16440305941.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.14.850875775193922429085936178692096.5 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{3}$ | ${\href{/padicField/17.9.0.1}{9} }^{2}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ | R | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{10}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.6a3.2 | $x^{6} + 4 x^{4} + 4 x^{3} + 7 x^{2} + 6 x + 5$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $$[2, 2]^{6}$$ |
| 2.6.2.12a3.2 | $x^{12} + 2 x^{10} + 4 x^{9} + x^{8} + 6 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 6 x + 3$ | $2$ | $6$ | $12$ | 12T105 | $$[2, 2, 2, 2]^{12}$$ | |
|
\(37\)
| 37.3.1.0a1.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 37.3.1.0a1.1 | $x^{3} + 6 x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 37.6.1.0a1.1 | $x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
| 37.2.3.4a1.3 | $x^{6} + 99 x^{5} + 3273 x^{4} + 36333 x^{3} + 6546 x^{2} + 544 x + 1303$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
|
\(229\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $12$ | $2$ | $6$ | $6$ |