Normalized defining polynomial
\( x^{18} - 30 x^{16} - 20 x^{15} + 360 x^{14} + 420 x^{13} - 3230 x^{12} - 7200 x^{11} + 8490 x^{10} + \cdots - 200 \)
Invariants
| Degree: | $18$ |
| |
| Signature: | $(6, 6)$ |
| |
| Discriminant: |
\(126949945835520000000000000000000\)
\(\medspace = 2^{34}\cdot 3^{18}\cdot 5^{19}\)
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| Root discriminant: | \(60.75\) |
| |
| Galois root discriminant: | $2^{7/3}3^{7/6}5^{71/60}\approx 121.94367865849007$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{40}a^{15}-\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{40}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{59\cdots 60}a^{17}+\frac{17\cdots 37}{59\cdots 60}a^{16}+\frac{36\cdots 99}{59\cdots 60}a^{15}+\frac{66\cdots 73}{59\cdots 16}a^{14}-\frac{12\cdots 47}{59\cdots 16}a^{13}-\frac{62\cdots 75}{59\cdots 16}a^{12}+\frac{16\cdots 19}{59\cdots 16}a^{11}+\frac{14\cdots 15}{59\cdots 16}a^{10}+\frac{24\cdots 16}{14\cdots 79}a^{9}-\frac{23\cdots 65}{59\cdots 16}a^{8}+\frac{20\cdots 97}{59\cdots 16}a^{7}+\frac{12\cdots 33}{47\cdots 09}a^{6}-\frac{40\cdots 89}{59\cdots 16}a^{5}+\frac{94\cdots 03}{59\cdots 16}a^{4}-\frac{55\cdots 57}{59\cdots 16}a^{3}-\frac{64\cdots 59}{14\cdots 79}a^{2}-\frac{58\cdots 67}{14\cdots 79}a-\frac{13\cdots 77}{29\cdots 58}$
| Monogenic: | Not computed | |
| 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: | Trivial group, which has order $1$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{18\cdots 97}{38\cdots 72}a^{17}-\frac{60\cdots 93}{19\cdots 36}a^{16}-\frac{28\cdots 33}{19\cdots 60}a^{15}-\frac{10\cdots 47}{19\cdots 36}a^{14}+\frac{17\cdots 11}{95\cdots 18}a^{13}+\frac{17\cdots 09}{19\cdots 36}a^{12}-\frac{32\cdots 53}{19\cdots 36}a^{11}-\frac{23\cdots 33}{95\cdots 18}a^{10}+\frac{28\cdots 53}{47\cdots 09}a^{9}+\frac{17\cdots 57}{95\cdots 18}a^{8}+\frac{60\cdots 45}{19\cdots 36}a^{7}+\frac{25\cdots 74}{47\cdots 09}a^{6}+\frac{96\cdots 59}{19\cdots 36}a^{5}+\frac{33\cdots 83}{95\cdots 18}a^{4}+\frac{22\cdots 91}{19\cdots 36}a^{3}-\frac{82\cdots 25}{95\cdots 18}a^{2}+\frac{10\cdots 05}{95\cdots 18}a-\frac{22\cdots 07}{95\cdots 18}$, $\frac{97\cdots 29}{19\cdots 36}a^{17}-\frac{50\cdots 15}{38\cdots 72}a^{16}-\frac{30\cdots 59}{19\cdots 36}a^{15}-\frac{23\cdots 08}{47\cdots 09}a^{14}+\frac{39\cdots 73}{19\cdots 36}a^{13}+\frac{13\cdots 31}{95\cdots 18}a^{12}-\frac{18\cdots 49}{95\cdots 18}a^{11}-\frac{59\cdots 49}{19\cdots 36}a^{10}+\frac{37\cdots 62}{47\cdots 09}a^{9}+\frac{42\cdots 43}{19\cdots 36}a^{8}+\frac{27\cdots 77}{95\cdots 18}a^{7}+\frac{22\cdots 54}{47\cdots 09}a^{6}+\frac{33\cdots 07}{95\cdots 18}a^{5}-\frac{28\cdots 81}{19\cdots 36}a^{4}-\frac{14\cdots 03}{95\cdots 18}a^{3}-\frac{27\cdots 55}{95\cdots 18}a^{2}-\frac{21\cdots 55}{95\cdots 18}a-\frac{19\cdots 72}{47\cdots 09}$, $\frac{54\cdots 89}{95\cdots 18}a^{17}-\frac{98\cdots 01}{47\cdots 09}a^{16}-\frac{65\cdots 37}{38\cdots 72}a^{15}-\frac{24\cdots 60}{47\cdots 09}a^{14}+\frac{20\cdots 37}{95\cdots 18}a^{13}+\frac{31\cdots 61}{19\cdots 36}a^{12}-\frac{93\cdots 02}{47\cdots 09}a^{11}-\frac{16\cdots 93}{47\cdots 09}a^{10}+\frac{12\cdots 85}{19\cdots 36}a^{9}+\frac{11\cdots 56}{47\cdots 09}a^{8}+\frac{77\cdots 81}{19\cdots 36}a^{7}+\frac{31\cdots 32}{47\cdots 09}a^{6}+\frac{31\cdots 75}{47\cdots 09}a^{5}+\frac{18\cdots 30}{47\cdots 09}a^{4}+\frac{13\cdots 25}{19\cdots 36}a^{3}-\frac{48\cdots 75}{47\cdots 09}a^{2}-\frac{66\cdots 75}{95\cdots 18}a-\frac{26\cdots 67}{95\cdots 18}$, $\frac{10\cdots 03}{11\cdots 32}a^{17}-\frac{12\cdots 37}{14\cdots 90}a^{16}-\frac{29\cdots 65}{11\cdots 32}a^{15}+\frac{14\cdots 03}{59\cdots 16}a^{14}+\frac{60\cdots 74}{14\cdots 79}a^{13}-\frac{17\cdots 95}{59\cdots 16}a^{12}-\frac{32\cdots 35}{59\cdots 16}a^{11}+\frac{36\cdots 55}{14\cdots 79}a^{10}+\frac{17\cdots 37}{29\cdots 58}a^{9}-\frac{10\cdots 36}{14\cdots 79}a^{8}-\frac{16\cdots 99}{59\cdots 16}a^{7}-\frac{19\cdots 67}{47\cdots 09}a^{6}-\frac{44\cdots 15}{59\cdots 16}a^{5}-\frac{11\cdots 90}{14\cdots 79}a^{4}-\frac{23\cdots 69}{59\cdots 16}a^{3}-\frac{50\cdots 97}{29\cdots 58}a^{2}-\frac{12\cdots 93}{29\cdots 58}a-\frac{13\cdots 31}{29\cdots 58}$, $\frac{37\cdots 71}{74\cdots 95}a^{17}+\frac{37\cdots 79}{29\cdots 80}a^{16}-\frac{19\cdots 95}{11\cdots 32}a^{15}-\frac{70\cdots 08}{14\cdots 79}a^{14}+\frac{29\cdots 69}{14\cdots 79}a^{13}+\frac{39\cdots 23}{59\cdots 16}a^{12}-\frac{23\cdots 88}{14\cdots 79}a^{11}-\frac{23\cdots 63}{29\cdots 58}a^{10}-\frac{75\cdots 05}{59\cdots 16}a^{9}+\frac{11\cdots 83}{29\cdots 58}a^{8}+\frac{49\cdots 83}{59\cdots 16}a^{7}+\frac{66\cdots 36}{47\cdots 09}a^{6}+\frac{28\cdots 89}{14\cdots 79}a^{5}+\frac{40\cdots 17}{29\cdots 58}a^{4}+\frac{25\cdots 31}{59\cdots 16}a^{3}+\frac{31\cdots 81}{14\cdots 79}a^{2}+\frac{23\cdots 97}{29\cdots 58}a+\frac{17\cdots 43}{29\cdots 58}$, $\frac{13\cdots 49}{14\cdots 90}a^{17}-\frac{45\cdots 75}{14\cdots 79}a^{16}-\frac{15\cdots 33}{59\cdots 60}a^{15}-\frac{15\cdots 10}{14\cdots 79}a^{14}+\frac{41\cdots 56}{14\cdots 79}a^{13}+\frac{17\cdots 45}{59\cdots 16}a^{12}-\frac{35\cdots 35}{14\cdots 79}a^{11}-\frac{81\cdots 68}{14\cdots 79}a^{10}+\frac{20\cdots 13}{59\cdots 16}a^{9}+\frac{44\cdots 65}{14\cdots 79}a^{8}+\frac{57\cdots 37}{59\cdots 16}a^{7}+\frac{82\cdots 24}{47\cdots 09}a^{6}+\frac{24\cdots 11}{14\cdots 79}a^{5}+\frac{18\cdots 91}{14\cdots 79}a^{4}+\frac{29\cdots 33}{59\cdots 16}a^{3}+\frac{30\cdots 40}{14\cdots 79}a^{2}+\frac{17\cdots 31}{29\cdots 58}a-\frac{80\cdots 07}{29\cdots 58}$, $\frac{11\cdots 79}{61\cdots 20}a^{17}+\frac{30\cdots 77}{24\cdots 08}a^{16}-\frac{74\cdots 65}{12\cdots 04}a^{15}-\frac{25\cdots 87}{61\cdots 02}a^{14}+\frac{64\cdots 49}{12\cdots 04}a^{13}+\frac{34\cdots 27}{61\cdots 02}a^{12}-\frac{12\cdots 01}{61\cdots 02}a^{11}-\frac{71\cdots 61}{12\cdots 04}a^{10}-\frac{20\cdots 27}{30\cdots 01}a^{9}+\frac{29\cdots 43}{12\cdots 04}a^{8}+\frac{44\cdots 83}{61\cdots 02}a^{7}+\frac{11\cdots 28}{99\cdots 71}a^{6}+\frac{10\cdots 17}{61\cdots 02}a^{5}+\frac{21\cdots 27}{12\cdots 04}a^{4}+\frac{58\cdots 63}{61\cdots 02}a^{3}+\frac{17\cdots 33}{61\cdots 02}a^{2}+\frac{81\cdots 31}{61\cdots 02}a+\frac{15\cdots 98}{30\cdots 01}$, $\frac{25\cdots 27}{59\cdots 60}a^{17}-\frac{19\cdots 61}{29\cdots 80}a^{16}-\frac{26\cdots 89}{59\cdots 60}a^{15}+\frac{11\cdots 21}{59\cdots 16}a^{14}+\frac{60\cdots 59}{29\cdots 58}a^{13}-\frac{13\cdots 19}{59\cdots 16}a^{12}-\frac{47\cdots 75}{59\cdots 16}a^{11}+\frac{63\cdots 45}{29\cdots 58}a^{10}+\frac{58\cdots 61}{29\cdots 58}a^{9}-\frac{29\cdots 17}{29\cdots 58}a^{8}-\frac{82\cdots 35}{59\cdots 16}a^{7}-\frac{52\cdots 53}{47\cdots 09}a^{6}-\frac{17\cdots 83}{59\cdots 16}a^{5}-\frac{30\cdots 93}{29\cdots 58}a^{4}+\frac{10\cdots 83}{59\cdots 16}a^{3}+\frac{11\cdots 01}{29\cdots 58}a^{2}-\frac{55\cdots 49}{29\cdots 58}a+\frac{62\cdots 49}{29\cdots 58}$, $\frac{21\cdots 49}{14\cdots 90}a^{17}-\frac{20\cdots 53}{59\cdots 60}a^{16}-\frac{26\cdots 79}{59\cdots 60}a^{15}-\frac{52\cdots 45}{29\cdots 58}a^{14}+\frac{33\cdots 75}{59\cdots 16}a^{13}+\frac{28\cdots 59}{59\cdots 16}a^{12}-\frac{77\cdots 36}{14\cdots 79}a^{11}-\frac{55\cdots 15}{59\cdots 16}a^{10}+\frac{11\cdots 87}{59\cdots 16}a^{9}+\frac{37\cdots 25}{59\cdots 16}a^{8}+\frac{57\cdots 07}{59\cdots 16}a^{7}+\frac{75\cdots 08}{47\cdots 09}a^{6}+\frac{25\cdots 36}{14\cdots 79}a^{5}+\frac{51\cdots 81}{59\cdots 16}a^{4}+\frac{15\cdots 43}{59\cdots 16}a^{3}+\frac{37\cdots 55}{29\cdots 58}a^{2}+\frac{66\cdots 04}{14\cdots 79}a+\frac{84\cdots 59}{29\cdots 58}$, $\frac{91\cdots 31}{59\cdots 60}a^{17}-\frac{14\cdots 47}{14\cdots 90}a^{16}-\frac{12\cdots 73}{29\cdots 80}a^{15}-\frac{40\cdots 29}{59\cdots 16}a^{14}+\frac{70\cdots 90}{14\cdots 79}a^{13}+\frac{11\cdots 03}{29\cdots 58}a^{12}-\frac{25\cdots 43}{59\cdots 16}a^{11}-\frac{12\cdots 49}{14\cdots 79}a^{10}+\frac{55\cdots 95}{59\cdots 16}a^{9}+\frac{74\cdots 07}{14\cdots 79}a^{8}+\frac{39\cdots 43}{29\cdots 58}a^{7}+\frac{10\cdots 67}{47\cdots 09}a^{6}+\frac{17\cdots 93}{59\cdots 16}a^{5}+\frac{43\cdots 79}{14\cdots 79}a^{4}+\frac{30\cdots 65}{14\cdots 79}a^{3}+\frac{28\cdots 21}{29\cdots 58}a^{2}+\frac{33\cdots 21}{14\cdots 79}a-\frac{16\cdots 92}{14\cdots 79}$, $\frac{33\cdots 97}{29\cdots 80}a^{17}+\frac{19\cdots 13}{59\cdots 60}a^{16}-\frac{38\cdots 11}{11\cdots 32}a^{15}-\frac{94\cdots 05}{29\cdots 58}a^{14}+\frac{21\cdots 59}{59\cdots 16}a^{13}+\frac{32\cdots 43}{59\cdots 16}a^{12}-\frac{89\cdots 75}{29\cdots 58}a^{11}-\frac{49\cdots 89}{59\cdots 16}a^{10}+\frac{17\cdots 37}{59\cdots 16}a^{9}+\frac{23\cdots 15}{59\cdots 16}a^{8}+\frac{72\cdots 99}{59\cdots 16}a^{7}+\frac{12\cdots 86}{47\cdots 09}a^{6}+\frac{11\cdots 11}{29\cdots 58}a^{5}+\frac{21\cdots 83}{59\cdots 16}a^{4}+\frac{13\cdots 19}{59\cdots 16}a^{3}+\frac{23\cdots 91}{29\cdots 58}a^{2}+\frac{19\cdots 13}{14\cdots 79}a+\frac{63\cdots 47}{29\cdots 58}$
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| Regulator: | \( 6503790952.27 \) (assuming GRH) |
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| Unit signature rank: | \( 6 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 6503790952.27 \cdot 1}{2\cdot\sqrt{126949945835520000000000000000000}}\cr\approx \mathstrut & 1.13652633137 \end{aligned}\] (assuming GRH)
Galois group
$S_3\times S_5$ (as 18T227):
| A non-solvable group of order 720 |
| The 21 conjugacy class representatives for $S_3\times S_5$ |
| Character table for $S_3\times S_5$ |
Intermediate fields
| 3.3.2700.1, 6.2.3200000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 15 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Degree 45 sibling: | data not computed |
| Minimal sibling: | 15.3.2938656153600000000000000000.1 |
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 | R | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | $15{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.6.10a1.5 | $x^{6} + 2 x^{5} + 4 x + 2$ | $6$ | $1$ | $10$ | $S_4$ | $$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$ |
| 2.1.12.24c1.14 | $x^{12} + 4 x^{8} + 2 x^{6} + 4 x^{5} + 4 x + 6$ | $12$ | $1$ | $24$ | $C_2 \times S_4$ | $$[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$$ | |
|
\(3\)
| 3.6.3.18a1.1 | $x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 26 x^{12} + 24 x^{11} + 39 x^{10} + 36 x^{9} + 54 x^{8} + 48 x^{7} + 67 x^{6} + 54 x^{5} + 54 x^{4} + 32 x^{3} + 42 x^{2} + 36 x + 23$ | $3$ | $6$ | $18$ | not computed | not computed |
|
\(5\)
| 5.1.3.2a1.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
| 5.1.15.17a1.4 | $x^{15} + 20 x^{3} + 5$ | $15$ | $1$ | $17$ | $F_5 \times S_3$ | $$[\frac{5}{4}]_{12}^{2}$$ |