Normalized defining polynomial
\( x^{18} - 7 x^{17} + 17 x^{16} + 17 x^{15} - 935 x^{14} + 799 x^{13} + 9231 x^{12} - 41463 x^{11} + \cdots + 113422599 \)
Invariants
| Degree: | $18$ |
| |
| Signature: | $(2, 8)$ |
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| Discriminant: |
\(40254497110927943179349807054456171205137\)
\(\medspace = 17^{33}\)
|
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| Root discriminant: | \(180.23\) |
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| Galois root discriminant: | $17^{543/272}\approx 286.0053344539552$ | ||
| Ramified primes: |
\(17\)
|
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| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}-\frac{1}{9}a^{5}-\frac{2}{9}a^{4}-\frac{4}{9}a^{3}-\frac{2}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{2}{9}a^{4}+\frac{2}{9}a^{3}-\frac{1}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{27}a^{13}-\frac{1}{27}a^{11}-\frac{1}{27}a^{10}+\frac{1}{27}a^{9}+\frac{1}{27}a^{8}-\frac{1}{27}a^{7}-\frac{1}{27}a^{6}+\frac{4}{27}a^{5}-\frac{1}{3}a^{4}-\frac{1}{27}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{27}a^{14}-\frac{1}{27}a^{12}-\frac{1}{27}a^{11}+\frac{1}{27}a^{10}+\frac{1}{27}a^{9}-\frac{1}{27}a^{8}-\frac{1}{27}a^{7}+\frac{4}{27}a^{6}-\frac{10}{27}a^{4}+\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{27}a^{15}-\frac{1}{27}a^{12}+\frac{1}{9}a^{7}-\frac{1}{27}a^{6}+\frac{1}{9}a^{5}-\frac{2}{9}a^{4}-\frac{1}{27}a^{3}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{16}-\frac{1}{27}a^{11}-\frac{1}{27}a^{10}+\frac{1}{27}a^{9}+\frac{4}{27}a^{8}-\frac{2}{27}a^{7}+\frac{2}{27}a^{6}-\frac{2}{27}a^{5}-\frac{10}{27}a^{4}-\frac{7}{27}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{70\cdots 37}a^{17}+\frac{33\cdots 81}{70\cdots 37}a^{16}+\frac{11\cdots 11}{70\cdots 37}a^{15}+\frac{63\cdots 48}{70\cdots 37}a^{14}-\frac{74\cdots 10}{70\cdots 37}a^{13}+\frac{26\cdots 00}{70\cdots 37}a^{12}+\frac{10\cdots 69}{87\cdots 77}a^{11}-\frac{12\cdots 35}{78\cdots 93}a^{10}+\frac{53\cdots 04}{78\cdots 93}a^{9}+\frac{48\cdots 93}{70\cdots 37}a^{8}+\frac{57\cdots 46}{70\cdots 37}a^{7}+\frac{18\cdots 23}{70\cdots 37}a^{6}+\frac{60\cdots 55}{70\cdots 37}a^{5}-\frac{34\cdots 51}{70\cdots 37}a^{4}-\frac{50\cdots 74}{70\cdots 37}a^{3}-\frac{39\cdots 75}{23\cdots 79}a^{2}+\frac{53\cdots 13}{26\cdots 31}a+\frac{18\cdots 57}{26\cdots 31}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{28\cdots 65}{70\cdots 37}a^{17}-\frac{30\cdots 07}{70\cdots 37}a^{16}+\frac{16\cdots 38}{70\cdots 37}a^{15}+\frac{21\cdots 13}{70\cdots 37}a^{14}-\frac{26\cdots 58}{70\cdots 37}a^{13}+\frac{17\cdots 96}{70\cdots 37}a^{12}+\frac{20\cdots 86}{78\cdots 93}a^{11}-\frac{44\cdots 28}{87\cdots 77}a^{10}+\frac{80\cdots 56}{78\cdots 93}a^{9}-\frac{25\cdots 88}{70\cdots 37}a^{8}-\frac{13\cdots 23}{70\cdots 37}a^{7}+\frac{12\cdots 13}{70\cdots 37}a^{6}-\frac{10\cdots 12}{70\cdots 37}a^{5}-\frac{56\cdots 68}{70\cdots 37}a^{4}+\frac{24\cdots 19}{70\cdots 37}a^{3}-\frac{31\cdots 89}{23\cdots 79}a^{2}+\frac{43\cdots 28}{26\cdots 31}a-\frac{40\cdots 57}{26\cdots 31}$, $\frac{14\cdots 08}{70\cdots 37}a^{17}-\frac{94\cdots 00}{70\cdots 37}a^{16}-\frac{26\cdots 52}{70\cdots 37}a^{15}+\frac{61\cdots 63}{70\cdots 37}a^{14}-\frac{51\cdots 15}{70\cdots 37}a^{13}+\frac{18\cdots 11}{70\cdots 37}a^{12}-\frac{23\cdots 48}{87\cdots 77}a^{11}-\frac{26\cdots 90}{26\cdots 31}a^{10}+\frac{64\cdots 56}{78\cdots 93}a^{9}-\frac{16\cdots 70}{70\cdots 37}a^{8}+\frac{25\cdots 70}{70\cdots 37}a^{7}+\frac{18\cdots 36}{70\cdots 37}a^{6}-\frac{16\cdots 83}{70\cdots 37}a^{5}+\frac{48\cdots 61}{70\cdots 37}a^{4}-\frac{80\cdots 68}{70\cdots 37}a^{3}+\frac{32\cdots 68}{23\cdots 79}a^{2}-\frac{26\cdots 36}{26\cdots 31}a+\frac{13\cdots 15}{26\cdots 31}$, $\frac{52\cdots 62}{70\cdots 37}a^{17}-\frac{23\cdots 05}{70\cdots 37}a^{16}+\frac{87\cdots 01}{70\cdots 37}a^{15}+\frac{14\cdots 78}{70\cdots 37}a^{14}-\frac{14\cdots 89}{70\cdots 37}a^{13}+\frac{14\cdots 58}{70\cdots 37}a^{12}+\frac{48\cdots 71}{78\cdots 93}a^{11}-\frac{29\cdots 58}{78\cdots 93}a^{10}-\frac{23\cdots 53}{78\cdots 93}a^{9}-\frac{15\cdots 02}{70\cdots 37}a^{8}-\frac{84\cdots 17}{70\cdots 37}a^{7}+\frac{61\cdots 47}{70\cdots 37}a^{6}-\frac{21\cdots 33}{70\cdots 37}a^{5}-\frac{39\cdots 30}{70\cdots 37}a^{4}+\frac{10\cdots 26}{70\cdots 37}a^{3}-\frac{93\cdots 80}{23\cdots 79}a^{2}+\frac{82\cdots 66}{26\cdots 31}a-\frac{78\cdots 76}{26\cdots 31}$, $\frac{14\cdots 68}{23\cdots 79}a^{17}-\frac{48\cdots 94}{23\cdots 79}a^{16}-\frac{76\cdots 26}{23\cdots 79}a^{15}+\frac{64\cdots 81}{23\cdots 79}a^{14}-\frac{10\cdots 42}{23\cdots 79}a^{13}-\frac{37\cdots 98}{23\cdots 79}a^{12}+\frac{39\cdots 64}{78\cdots 93}a^{11}+\frac{26\cdots 80}{78\cdots 93}a^{10}+\frac{39\cdots 83}{78\cdots 93}a^{9}+\frac{99\cdots 25}{23\cdots 79}a^{8}+\frac{23\cdots 35}{23\cdots 79}a^{7}+\frac{77\cdots 79}{23\cdots 79}a^{6}+\frac{13\cdots 18}{23\cdots 79}a^{5}+\frac{16\cdots 44}{23\cdots 79}a^{4}+\frac{37\cdots 67}{23\cdots 79}a^{3}-\frac{24\cdots 29}{78\cdots 93}a^{2}+\frac{56\cdots 03}{29\cdots 59}a-\frac{38\cdots 22}{87\cdots 77}$, $\frac{16\cdots 84}{70\cdots 37}a^{17}-\frac{10\cdots 01}{70\cdots 37}a^{16}+\frac{19\cdots 39}{70\cdots 37}a^{15}+\frac{49\cdots 15}{70\cdots 37}a^{14}-\frac{15\cdots 48}{70\cdots 37}a^{13}+\frac{43\cdots 47}{70\cdots 37}a^{12}+\frac{21\cdots 02}{87\cdots 77}a^{11}-\frac{61\cdots 60}{78\cdots 93}a^{10}+\frac{29\cdots 97}{78\cdots 93}a^{9}+\frac{65\cdots 20}{70\cdots 37}a^{8}-\frac{60\cdots 94}{70\cdots 37}a^{7}+\frac{81\cdots 88}{70\cdots 37}a^{6}-\frac{41\cdots 46}{70\cdots 37}a^{5}-\frac{12\cdots 74}{70\cdots 37}a^{4}+\frac{44\cdots 99}{70\cdots 37}a^{3}-\frac{51\cdots 20}{23\cdots 79}a^{2}+\frac{42\cdots 67}{26\cdots 31}a-\frac{51\cdots 32}{26\cdots 31}$, $\frac{13\cdots 79}{23\cdots 79}a^{17}-\frac{75\cdots 09}{23\cdots 79}a^{16}+\frac{63\cdots 24}{23\cdots 79}a^{15}+\frac{71\cdots 23}{23\cdots 79}a^{14}-\frac{12\cdots 61}{23\cdots 79}a^{13}-\frac{86\cdots 10}{23\cdots 79}a^{12}+\frac{21\cdots 91}{29\cdots 59}a^{11}-\frac{11\cdots 30}{78\cdots 93}a^{10}+\frac{15\cdots 41}{26\cdots 31}a^{9}+\frac{79\cdots 60}{23\cdots 79}a^{8}-\frac{25\cdots 56}{23\cdots 79}a^{7}+\frac{49\cdots 24}{23\cdots 79}a^{6}+\frac{51\cdots 99}{23\cdots 79}a^{5}-\frac{14\cdots 82}{23\cdots 79}a^{4}+\frac{38\cdots 94}{23\cdots 79}a^{3}-\frac{12\cdots 56}{78\cdots 93}a^{2}+\frac{23\cdots 30}{29\cdots 59}a+\frac{14\cdots 06}{87\cdots 77}$, $\frac{17\cdots 87}{23\cdots 79}a^{17}-\frac{18\cdots 11}{23\cdots 79}a^{16}+\frac{97\cdots 02}{23\cdots 79}a^{15}-\frac{31\cdots 91}{23\cdots 79}a^{14}-\frac{58\cdots 71}{23\cdots 79}a^{13}+\frac{40\cdots 72}{23\cdots 79}a^{12}+\frac{11\cdots 58}{29\cdots 59}a^{11}-\frac{83\cdots 36}{26\cdots 31}a^{10}+\frac{69\cdots 17}{26\cdots 31}a^{9}-\frac{17\cdots 07}{23\cdots 79}a^{8}+\frac{52\cdots 75}{23\cdots 79}a^{7}-\frac{57\cdots 22}{23\cdots 79}a^{6}+\frac{37\cdots 66}{23\cdots 79}a^{5}+\frac{18\cdots 31}{23\cdots 79}a^{4}-\frac{43\cdots 10}{23\cdots 79}a^{3}+\frac{26\cdots 71}{78\cdots 93}a^{2}-\frac{22\cdots 59}{87\cdots 77}a+\frac{16\cdots 65}{87\cdots 77}$, $\frac{17\cdots 62}{70\cdots 37}a^{17}-\frac{19\cdots 36}{70\cdots 37}a^{16}+\frac{71\cdots 14}{70\cdots 37}a^{15}+\frac{40\cdots 65}{70\cdots 37}a^{14}-\frac{22\cdots 44}{70\cdots 37}a^{13}+\frac{82\cdots 25}{70\cdots 37}a^{12}+\frac{28\cdots 47}{78\cdots 93}a^{11}-\frac{22\cdots 54}{78\cdots 93}a^{10}+\frac{17\cdots 52}{26\cdots 31}a^{9}+\frac{47\cdots 46}{70\cdots 37}a^{8}-\frac{63\cdots 90}{70\cdots 37}a^{7}+\frac{10\cdots 84}{70\cdots 37}a^{6}-\frac{84\cdots 30}{70\cdots 37}a^{5}-\frac{42\cdots 57}{70\cdots 37}a^{4}+\frac{12\cdots 12}{70\cdots 37}a^{3}-\frac{70\cdots 28}{23\cdots 79}a^{2}+\frac{65\cdots 89}{26\cdots 31}a-\frac{43\cdots 26}{26\cdots 31}$, $\frac{39\cdots 01}{70\cdots 37}a^{17}-\frac{14\cdots 09}{70\cdots 37}a^{16}+\frac{28\cdots 88}{70\cdots 37}a^{15}+\frac{22\cdots 83}{70\cdots 37}a^{14}-\frac{31\cdots 64}{70\cdots 37}a^{13}-\frac{67\cdots 28}{70\cdots 37}a^{12}+\frac{15\cdots 35}{78\cdots 93}a^{11}-\frac{22\cdots 56}{78\cdots 93}a^{10}-\frac{17\cdots 47}{78\cdots 93}a^{9}+\frac{45\cdots 73}{70\cdots 37}a^{8}-\frac{41\cdots 41}{70\cdots 37}a^{7}+\frac{31\cdots 34}{70\cdots 37}a^{6}+\frac{93\cdots 50}{70\cdots 37}a^{5}-\frac{30\cdots 90}{70\cdots 37}a^{4}+\frac{60\cdots 77}{70\cdots 37}a^{3}-\frac{33\cdots 17}{23\cdots 79}a^{2}+\frac{27\cdots 78}{26\cdots 31}a-\frac{19\cdots 36}{26\cdots 31}$
|
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| Regulator: | \( 45413380405400 \) (assuming GRH) |
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| Unit signature rank: | \( 2 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 45413380405400 \cdot 1}{2\cdot\sqrt{40254497110927943179349807054456171205137}}\cr\approx \mathstrut & 1.09962744387092 \end{aligned}\] (assuming GRH)
Galois group
$\PGL(2,17)$ (as 18T468):
| A non-solvable group of order 4896 |
| The 19 conjugacy class representatives for $\PGL(2,17)$ |
| Character table for $\PGL(2,17)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 36 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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{9}$ | $16{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $18$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | $18$ | $16{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 17.1.17.33a1.11 | $x^{17} + 2890 x + 17$ | $17$ | $1$ | $33$ | $F_{17}$ | $$[\frac{33}{16}]_{16}$$ |
Spectrum of ring of integers
Additional information
This field is associated with the 17-division points on any elliptic curve in the isogeny class 17.a1.