Normalized defining polynomial
\( x^{15} - 4 x^{14} - 6 x^{13} + 24 x^{12} + 1117 x^{11} + 780 x^{10} + 2624 x^{9} - 648 x^{8} + \cdots - 16384 \)
Invariants
| Degree: | $15$ |
| |
| Signature: | $[3, 6]$ |
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| Discriminant: |
\(1744064843158258782713344097088061\)
\(\medspace = 229^{5}\cdot 120721^{2}\cdot 435923^{2}\)
|
| |
| Root discriminant: | \(164.48\) |
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| Galois root discriminant: | $229^{1/2}120721^{1/2}435923^{1/2}\approx 3471475.0252028317$ | ||
| Ramified primes: |
\(229\), \(120721\), \(435923\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{229}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{6}+\frac{1}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{7}-\frac{1}{16}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{32}a^{8}-\frac{1}{32}a^{7}-\frac{1}{32}a^{6}+\frac{1}{32}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{64}a^{9}-\frac{1}{64}a^{8}-\frac{1}{64}a^{7}+\frac{1}{64}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}$, $\frac{1}{64}a^{10}-\frac{1}{32}a^{7}-\frac{1}{64}a^{6}+\frac{1}{32}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{896}a^{11}-\frac{1}{448}a^{10}+\frac{1}{448}a^{9}+\frac{25}{896}a^{7}+\frac{9}{448}a^{6}+\frac{3}{224}a^{5}-\frac{11}{112}a^{4}-\frac{1}{8}a^{3}-\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{3}{7}$, $\frac{1}{3584}a^{12}-\frac{1}{1792}a^{11}-\frac{13}{1792}a^{10}-\frac{1}{128}a^{9}+\frac{53}{3584}a^{8}+\frac{51}{1792}a^{7}+\frac{3}{896}a^{6}-\frac{1}{112}a^{5}-\frac{1}{16}a^{4}+\frac{3}{112}a^{3}+\frac{3}{56}a^{2}-\frac{11}{28}a$, $\frac{1}{14336}a^{13}-\frac{1}{7168}a^{12}-\frac{1}{7168}a^{11}+\frac{9}{3584}a^{10}-\frac{11}{14336}a^{9}-\frac{5}{7168}a^{8}-\frac{99}{3584}a^{7}+\frac{25}{896}a^{6}-\frac{19}{448}a^{5}-\frac{1}{64}a^{4}+\frac{17}{224}a^{3}-\frac{1}{16}a^{2}+\frac{9}{28}a+\frac{1}{7}$, $\frac{1}{19\cdots 12}a^{14}-\frac{17\cdots 57}{99\cdots 56}a^{13}+\frac{98\cdots 43}{99\cdots 56}a^{12}-\frac{11\cdots 39}{49\cdots 28}a^{11}+\frac{13\cdots 13}{19\cdots 12}a^{10}+\frac{62\cdots 51}{99\cdots 56}a^{9}-\frac{11\cdots 03}{90\cdots 24}a^{8}-\frac{19\cdots 35}{44\cdots 44}a^{7}-\frac{24\cdots 27}{96\cdots 94}a^{6}+\frac{52\cdots 39}{88\cdots 88}a^{5}-\frac{33\cdots 75}{31\cdots 08}a^{4}-\frac{15\cdots 81}{15\cdots 04}a^{3}+\frac{34\cdots 37}{48\cdots 47}a^{2}-\frac{22\cdots 09}{19\cdots 88}a-\frac{20\cdots 22}{69\cdots 21}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ (assuming GRH) |
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| Narrow class group: | $C_{6}$, which has order $6$ (assuming GRH) |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{17\cdots 67}{35\cdots 52}a^{14}-\frac{35\cdots 33}{17\cdots 76}a^{13}-\frac{48\cdots 13}{17\cdots 76}a^{12}+\frac{52\cdots 63}{44\cdots 44}a^{11}+\frac{19\cdots 47}{35\cdots 52}a^{10}+\frac{59\cdots 75}{17\cdots 76}a^{9}+\frac{10\cdots 25}{81\cdots 52}a^{8}-\frac{38\cdots 69}{88\cdots 88}a^{7}+\frac{27\cdots 27}{22\cdots 72}a^{6}+\frac{77\cdots 81}{11\cdots 36}a^{5}+\frac{82\cdots 01}{27\cdots 84}a^{4}+\frac{40\cdots 53}{55\cdots 68}a^{3}+\frac{54\cdots 68}{69\cdots 21}a^{2}+\frac{43\cdots 54}{69\cdots 21}a-\frac{13\cdots 77}{69\cdots 21}$, $\frac{86\cdots 89}{17\cdots 76}a^{14}-\frac{73\cdots 47}{35\cdots 52}a^{13}-\frac{42\cdots 87}{17\cdots 76}a^{12}+\frac{21\cdots 21}{17\cdots 76}a^{11}+\frac{95\cdots 91}{17\cdots 76}a^{10}+\frac{88\cdots 69}{35\cdots 52}a^{9}+\frac{39\cdots 21}{32\cdots 08}a^{8}-\frac{43\cdots 83}{69\cdots 21}a^{7}+\frac{54\cdots 47}{44\cdots 44}a^{6}+\frac{56\cdots 69}{11\cdots 36}a^{5}+\frac{16\cdots 69}{55\cdots 68}a^{4}-\frac{20\cdots 87}{55\cdots 68}a^{3}+\frac{54\cdots 02}{69\cdots 21}a^{2}-\frac{43\cdots 90}{69\cdots 21}a+\frac{24\cdots 91}{69\cdots 21}$, $\frac{62\cdots 01}{99\cdots 56}a^{14}-\frac{88\cdots 31}{24\cdots 64}a^{13}-\frac{80\cdots 25}{49\cdots 28}a^{12}+\frac{29\cdots 89}{62\cdots 16}a^{11}+\frac{54\cdots 05}{99\cdots 56}a^{10}-\frac{12\cdots 55}{24\cdots 64}a^{9}-\frac{22\cdots 47}{45\cdots 12}a^{8}+\frac{35\cdots 71}{31\cdots 08}a^{7}+\frac{71\cdots 27}{62\cdots 16}a^{6}-\frac{24\cdots 55}{31\cdots 08}a^{5}-\frac{61\cdots 71}{38\cdots 76}a^{4}+\frac{28\cdots 31}{11\cdots 36}a^{3}+\frac{14\cdots 07}{27\cdots 84}a^{2}-\frac{14\cdots 61}{19\cdots 88}a-\frac{28\cdots 04}{48\cdots 47}$, $\frac{37\cdots 27}{99\cdots 56}a^{14}-\frac{72\cdots 03}{49\cdots 28}a^{13}-\frac{12\cdots 53}{49\cdots 28}a^{12}+\frac{21\cdots 81}{24\cdots 64}a^{11}+\frac{42\cdots 99}{99\cdots 56}a^{10}+\frac{17\cdots 53}{49\cdots 28}a^{9}+\frac{21\cdots 05}{22\cdots 56}a^{8}-\frac{74\cdots 97}{12\cdots 32}a^{7}+\frac{58\cdots 73}{62\cdots 16}a^{6}+\frac{23\cdots 15}{31\cdots 08}a^{5}+\frac{36\cdots 01}{15\cdots 04}a^{4}-\frac{35\cdots 23}{38\cdots 76}a^{3}+\frac{22\cdots 73}{38\cdots 76}a^{2}+\frac{47\cdots 57}{27\cdots 84}a+\frac{21\cdots 71}{48\cdots 47}$, $\frac{25\cdots 27}{19\cdots 12}a^{14}-\frac{42\cdots 71}{35\cdots 52}a^{13}+\frac{37\cdots 63}{99\cdots 56}a^{12}-\frac{24\cdots 37}{12\cdots 32}a^{11}+\frac{23\cdots 71}{19\cdots 12}a^{10}-\frac{12\cdots 41}{24\cdots 64}a^{9}+\frac{28\cdots 59}{22\cdots 56}a^{8}-\frac{33\cdots 43}{24\cdots 64}a^{7}+\frac{88\cdots 29}{31\cdots 08}a^{6}-\frac{73\cdots 55}{62\cdots 16}a^{5}+\frac{44\cdots 75}{15\cdots 04}a^{4}-\frac{11\cdots 65}{77\cdots 52}a^{3}-\frac{32\cdots 39}{77\cdots 52}a^{2}+\frac{11\cdots 09}{19\cdots 88}a+\frac{99\cdots 81}{48\cdots 47}$, $\frac{22\cdots 27}{99\cdots 56}a^{14}+\frac{79\cdots 39}{24\cdots 64}a^{13}-\frac{41\cdots 97}{70\cdots 04}a^{12}-\frac{97\cdots 33}{15\cdots 04}a^{11}+\frac{16\cdots 07}{99\cdots 56}a^{10}+\frac{12\cdots 35}{24\cdots 64}a^{9}+\frac{84\cdots 37}{45\cdots 12}a^{8}+\frac{17\cdots 77}{62\cdots 16}a^{7}-\frac{10\cdots 83}{62\cdots 16}a^{6}+\frac{18\cdots 89}{31\cdots 08}a^{5}+\frac{27\cdots 69}{96\cdots 94}a^{4}+\frac{50\cdots 17}{77\cdots 52}a^{3}+\frac{17\cdots 49}{19\cdots 88}a^{2}-\frac{54\cdots 27}{27\cdots 84}a-\frac{42\cdots 09}{69\cdots 21}$, $\frac{87\cdots 75}{19\cdots 12}a^{14}-\frac{42\cdots 61}{12\cdots 32}a^{13}-\frac{49\cdots 29}{99\cdots 56}a^{12}+\frac{28\cdots 75}{12\cdots 32}a^{11}+\frac{12\cdots 35}{19\cdots 12}a^{10}-\frac{15\cdots 75}{12\cdots 32}a^{9}-\frac{46\cdots 41}{45\cdots 12}a^{8}-\frac{17\cdots 31}{35\cdots 52}a^{7}+\frac{78\cdots 55}{62\cdots 16}a^{6}-\frac{18\cdots 51}{62\cdots 16}a^{5}-\frac{12\cdots 89}{15\cdots 04}a^{4}-\frac{50\cdots 23}{13\cdots 42}a^{3}+\frac{61\cdots 53}{11\cdots 36}a^{2}-\frac{23\cdots 64}{48\cdots 47}a-\frac{88\cdots 56}{48\cdots 47}$, $\frac{28\cdots 87}{19\cdots 12}a^{14}-\frac{39\cdots 51}{70\cdots 04}a^{13}-\frac{93\cdots 09}{99\cdots 56}a^{12}+\frac{83\cdots 81}{24\cdots 64}a^{11}+\frac{46\cdots 05}{28\cdots 16}a^{10}+\frac{66\cdots 63}{49\cdots 28}a^{9}+\frac{18\cdots 27}{45\cdots 12}a^{8}-\frac{10\cdots 85}{24\cdots 64}a^{7}+\frac{11\cdots 11}{31\cdots 08}a^{6}+\frac{17\cdots 61}{62\cdots 16}a^{5}+\frac{36\cdots 65}{38\cdots 76}a^{4}+\frac{88\cdots 13}{38\cdots 76}a^{3}+\frac{18\cdots 47}{77\cdots 52}a^{2}+\frac{13\cdots 23}{19\cdots 88}a+\frac{85\cdots 09}{48\cdots 47}$
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| Regulator: | \( 60181699125.0 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 60181699125.0 \cdot 3}{2\cdot\sqrt{1744064843158258782713344097088061}}\cr\approx \mathstrut & 1.06400480074 \end{aligned}\] (assuming GRH)
Galois group
$A_5\wr S_3$ (as 15T96):
| A non-solvable group of order 1296000 |
| The 65 conjugacy class representatives for $A_5\wr S_3$ |
| Character table for $A_5\wr S_3$ |
Intermediate fields
| 3.3.229.1 |
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 30 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Degree 45 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 | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $15$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{5}$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $15$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | $15$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.5.0.1}{5} }^{3}$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(229\)
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
|
\(120721\)
| $\Q_{120721}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $$[\ ]^{10}$$ | ||
|
\(435923\)
| $\Q_{435923}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |