Normalized defining polynomial
\( x^{20} - 20 x^{18} + 170 x^{16} - x^{15} - 800 x^{14} + 15 x^{13} + 2275 x^{12} - 90 x^{11} - 4004 x^{10} + \cdots + 1 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(20, 0)$ |
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| Discriminant: |
\(171855208463966846466064453125\)
\(\medspace = 3^{10}\cdot 5^{35}\)
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| Root discriminant: | \(28.96\) |
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| Galois root discriminant: | $3^{1/2}5^{7/4}\approx 28.957304632206725$ | ||
| Ramified primes: |
\(3\), \(5\)
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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: | \(75=3\cdot 5^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{75}(64,·)$, $\chi_{75}(1,·)$, $\chi_{75}(2,·)$, $\chi_{75}(4,·)$, $\chi_{75}(8,·)$, $\chi_{75}(16,·)$, $\chi_{75}(17,·)$, $\chi_{75}(19,·)$, $\chi_{75}(23,·)$, $\chi_{75}(68,·)$, $\chi_{75}(31,·)$, $\chi_{75}(32,·)$, $\chi_{75}(34,·)$, $\chi_{75}(38,·)$, $\chi_{75}(46,·)$, $\chi_{75}(47,·)$, $\chi_{75}(49,·)$, $\chi_{75}(53,·)$, $\chi_{75}(61,·)$, $\chi_{75}(62,·)$$\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: | $C_{2}$, which has order $2$ (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: |
$a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{19}-19a^{17}-a^{16}+152a^{15}+15a^{14}-665a^{13}-90a^{12}+1729a^{11}+275a^{10}-2717a^{9}-450a^{8}+2508a^{7}+379a^{6}-1254a^{5}-145a^{4}+285a^{3}+20a^{2}-20a$, $a^{14}-14a^{12}+a^{11}+77a^{10}-11a^{9}-210a^{8}+44a^{7}+294a^{6}-77a^{5}-196a^{4}+55a^{3}+49a^{2}-11a-2$, $a^{17}-16a^{15}-a^{14}+104a^{13}+13a^{12}-353a^{11}-65a^{10}+671a^{9}+157a^{8}-716a^{7}-189a^{6}+413a^{5}+105a^{4}-119a^{3}-20a^{2}+13a$, $a^{3}-3a$, $a^{18}-18a^{16}+a^{15}+135a^{14}-15a^{13}-545a^{12}+90a^{11}+1275a^{10}-274a^{9}-1728a^{8}+441a^{7}+1275a^{6}-351a^{5}-441a^{4}+111a^{3}+54a^{2}-9a-1$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{17}-17a^{15}+119a^{13}-a^{12}-442a^{11}+12a^{10}+936a^{9}-53a^{8}-1131a^{7}+104a^{6}+741a^{5}-85a^{4}-235a^{3}+20a^{2}+29a+1$, $a^{18}-19a^{16}+a^{15}+151a^{14}-16a^{13}-649a^{12}+103a^{11}+1626a^{10}-339a^{9}-2378a^{8}+596a^{7}+1913a^{6}-527a^{5}-734a^{4}+194a^{3}+105a^{2}-24a-2$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{4}-4a^{2}+2$, $a^{12}-a^{11}-11a^{10}+10a^{9}+45a^{8}-36a^{7}-84a^{6}+56a^{5}+70a^{4}-35a^{3}-21a^{2}+6a+1$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+661a^{8}-680a^{6}+356a^{4}-80a^{2}+5$, $a$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-3$, $a^{2}-2$, $a^{18}-a^{17}-18a^{16}+17a^{15}+135a^{14}-120a^{13}-546a^{12}+455a^{11}+1287a^{10}-1000a^{9}-1782a^{8}+1279a^{7}+1386a^{6}-903a^{5}-540a^{4}+310a^{3}+80a^{2}-40a$
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| Regulator: | \( 122630529.842 \) (assuming GRH) |
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| Unit signature rank: | \( 19 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 122630529.842 \cdot 1}{2\cdot\sqrt{171855208463966846466064453125}}\cr\approx \mathstrut & 0.155091205907 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 5.5.390625.1, \(\Q(\zeta_{25})^+\) |
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 | $20$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
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\(3\)
| 3.10.2.10a1.1 | $x^{20} + 4 x^{16} + 4 x^{15} + 4 x^{14} + 4 x^{12} + 10 x^{11} + 16 x^{10} + 8 x^{9} + 4 x^{8} + 4 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + x^{2} + 7 x + 4$ | $2$ | $10$ | $10$ | 20T1 | $$[\ ]_{2}^{10}$$ |
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\(5\)
| 5.1.20.35a1.500 | $x^{20} + 5 x^{16} + 120$ | $20$ | $1$ | $35$ | 20T1 | not computed |