Normalized defining polynomial
\( x^{20} - x^{19} - 20 x^{18} + 19 x^{17} + 170 x^{16} - 151 x^{15} - 801 x^{14} + 650 x^{13} + 2289 x^{12} + \cdots + 1 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(20, 0)$ |
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| Discriminant: |
\(169675210983039290802001953125\)
\(\medspace = 5^{15}\cdot 11^{18}\)
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| Root discriminant: | \(28.94\) |
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| Galois root discriminant: | $5^{3/4}11^{9/10}\approx 28.93882675684651$ | ||
| Ramified primes: |
\(5\), \(11\)
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| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(55=5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{55}(1,·)$, $\chi_{55}(2,·)$, $\chi_{55}(4,·)$, $\chi_{55}(7,·)$, $\chi_{55}(8,·)$, $\chi_{55}(9,·)$, $\chi_{55}(13,·)$, $\chi_{55}(14,·)$, $\chi_{55}(16,·)$, $\chi_{55}(17,·)$, $\chi_{55}(18,·)$, $\chi_{55}(26,·)$, $\chi_{55}(28,·)$, $\chi_{55}(31,·)$, $\chi_{55}(32,·)$, $\chi_{55}(34,·)$, $\chi_{55}(36,·)$, $\chi_{55}(43,·)$, $\chi_{55}(49,·)$, $\chi_{55}(52,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
| 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: | $19$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{19}-19a^{17}+151a^{15}+a^{14}-650a^{13}-14a^{12}+1639a^{11}+77a^{10}-2442a^{9}-209a^{8}+2058a^{7}+286a^{6}-877a^{5}-176a^{4}+151a^{3}+33a^{2}-12a$, $a^{10}-10a^{8}+35a^{6}+a^{5}-50a^{4}-5a^{3}+25a^{2}+5a-1$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-377a^{5}+135a^{3}-10a$, $a^{5}-5a^{3}+5a$, $a^{11}-11a^{9}+44a^{7}-76a^{5}+50a^{3}-6a+1$, $a^{19}-19a^{17}+151a^{15}+a^{14}-649a^{13}-14a^{12}+1626a^{11}+77a^{10}-2377a^{9}-209a^{8}+1902a^{7}+286a^{6}-694a^{5}-176a^{4}+55a^{3}+34a^{2}+6a-3$, $a^{18}-18a^{16}+a^{15}+135a^{14}-15a^{13}-546a^{12}+90a^{11}+1287a^{10}-275a^{9}-1782a^{8}+451a^{7}+1386a^{6}-385a^{5}-539a^{4}+154a^{3}+77a^{2}-22a$, $a^{10}-11a^{8}+43a^{6}+a^{5}-70a^{4}-6a^{3}+41a^{2}+8a-3$, $a^{3}-3a$, $a^{4}-5a^{2}+5$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{15}-15a^{13}+91a^{11}-286a^{9}+494a^{7}-455a^{5}+195a^{3}-26a+1$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{2}-2$, $a^{2}-1$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{16}-a^{15}-15a^{14}+14a^{13}+91a^{12}-78a^{11}-286a^{10}+220a^{9}+495a^{8}-330a^{7}-462a^{6}+253a^{5}+209a^{4}-88a^{3}-33a^{2}+11a$, $a^{17}-17a^{15}+119a^{13}-443a^{11}+946a^{9}-1166a^{7}+791a^{5}-259a^{3}+28a$
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| Regulator: | \( 115949178.41 \) (assuming GRH) |
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| Unit signature rank: | \( 18 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 115949178.41 \cdot 1}{2\cdot\sqrt{169675210983039290802001953125}}\cr\approx \mathstrut & 0.14758030187 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{110 -22 \sqrt{5}})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | $20$ | R | $20$ | R | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.5.4.15a1.4 | $x^{20} + 16 x^{16} + 12 x^{15} + 96 x^{12} + 144 x^{11} + 54 x^{10} + 256 x^{8} + 576 x^{7} + 432 x^{6} + 108 x^{5} + 256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 86$ | $4$ | $5$ | $15$ | 20T1 | not computed |
|
\(11\)
| 11.1.10.9a1.1 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |
| 11.1.10.9a1.1 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |