Normalized defining polynomial
\( x^{20} - 120 x^{18} + 6120 x^{16} - 172800 x^{14} + 2948400 x^{12} - 31142880 x^{10} + 200620800 x^{8} + \cdots + 302330880 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(20, 0)$ |
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| Discriminant: |
\(184528125000000000000000000000000000000\)
\(\medspace = 2^{30}\cdot 3^{10}\cdot 5^{35}\)
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| Root discriminant: | \(81.90\) |
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| Galois root discriminant: | $2^{3/2}3^{1/2}5^{7/4}\approx 81.903625881272$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
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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: | \(600=2^{3}\cdot 3\cdot 5^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{600}(1,·)$, $\chi_{600}(197,·)$, $\chi_{600}(481,·)$, $\chi_{600}(77,·)$, $\chi_{600}(173,·)$, $\chi_{600}(529,·)$, $\chi_{600}(533,·)$, $\chi_{600}(409,·)$, $\chi_{600}(413,·)$, $\chi_{600}(289,·)$, $\chi_{600}(293,·)$, $\chi_{600}(49,·)$, $\chi_{600}(169,·)$, $\chi_{600}(557,·)$, $\chi_{600}(241,·)$, $\chi_{600}(437,·)$, $\chi_{600}(361,·)$, $\chi_{600}(121,·)$, $\chi_{600}(317,·)$, $\chi_{600}(53,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{6}a^{2}$, $\frac{1}{6}a^{3}$, $\frac{1}{36}a^{4}$, $\frac{1}{36}a^{5}$, $\frac{1}{216}a^{6}$, $\frac{1}{216}a^{7}$, $\frac{1}{1296}a^{8}$, $\frac{1}{1296}a^{9}$, $\frac{1}{7776}a^{10}$, $\frac{1}{7776}a^{11}$, $\frac{1}{46656}a^{12}$, $\frac{1}{46656}a^{13}$, $\frac{1}{279936}a^{14}$, $\frac{1}{279936}a^{15}$, $\frac{1}{1679616}a^{16}$, $\frac{1}{1679616}a^{17}$, $\frac{1}{10077696}a^{18}$, $\frac{1}{10077696}a^{19}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (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: |
$\frac{1}{7776}a^{10}-\frac{5}{648}a^{8}+\frac{35}{216}a^{6}-\frac{25}{18}a^{4}+\frac{25}{6}a^{2}-2$, $\frac{1}{36}a^{4}-\frac{2}{3}a^{2}+2$, $\frac{1}{1296}a^{8}-\frac{1}{27}a^{6}+\frac{5}{9}a^{4}-\frac{8}{3}a^{2}+2$, $\frac{1}{6}a^{2}-2$, $\frac{1}{279936}a^{14}-\frac{13}{46656}a^{12}+\frac{11}{1296}a^{10}-\frac{55}{432}a^{8}+\frac{35}{36}a^{6}-\frac{127}{36}a^{4}+\frac{31}{6}a^{2}-2$, $\frac{1}{10077696}a^{18}-\frac{17}{1679616}a^{16}+\frac{59}{139968}a^{14}-\frac{143}{15552}a^{12}+\frac{871}{7776}a^{10}-\frac{61}{81}a^{8}+\frac{283}{108}a^{6}-\frac{157}{36}a^{4}+\frac{7}{2}a^{2}-2$, $\frac{1}{1679616}a^{16}-\frac{1}{17496}a^{14}+\frac{13}{5832}a^{12}-\frac{353}{7776}a^{10}+\frac{223}{432}a^{8}-\frac{175}{54}a^{6}+\frac{31}{3}a^{4}-\frac{27}{2}a^{2}+3$, $\frac{1}{46656}a^{12}-\frac{1}{648}a^{10}+\frac{1}{24}a^{8}-\frac{14}{27}a^{6}+\frac{35}{12}a^{4}-6a^{2}+2$, $\frac{1}{216}a^{6}-\frac{1}{6}a^{4}+\frac{3}{2}a^{2}-2$, $\frac{1}{279936}a^{14}-\frac{7}{23328}a^{12}+\frac{77}{7776}a^{10}-\frac{35}{216}a^{8}+\frac{1}{216}a^{7}+\frac{49}{36}a^{6}-\frac{7}{36}a^{5}-\frac{49}{9}a^{4}+\frac{7}{3}a^{3}+\frac{49}{6}a^{2}-7a+1$, $\frac{1}{10077696}a^{18}-\frac{1}{93312}a^{16}+\frac{5}{10368}a^{14}-\frac{547}{46656}a^{12}+\frac{1}{7776}a^{11}+\frac{433}{2592}a^{10}-\frac{11}{1296}a^{9}-\frac{1837}{1296}a^{8}+\frac{11}{54}a^{7}+\frac{251}{36}a^{6}-\frac{77}{36}a^{5}-\frac{665}{36}a^{4}+\frac{55}{6}a^{3}+\frac{67}{3}a^{2}-11a-11$, $\frac{1}{10077696}a^{18}-\frac{1}{93312}a^{16}+\frac{5}{10368}a^{14}-\frac{91}{7776}a^{12}+\frac{143}{864}a^{10}-\frac{1}{1296}a^{9}-\frac{11}{8}a^{8}+\frac{1}{24}a^{7}+\frac{77}{12}a^{6}-\frac{3}{4}a^{5}-15a^{4}+5a^{3}+\frac{27}{2}a^{2}-9a+1$, $\frac{1}{1679616}a^{17}-\frac{1}{1679616}a^{16}-\frac{17}{279936}a^{15}+\frac{1}{17496}a^{14}+\frac{119}{46656}a^{13}-\frac{13}{5832}a^{12}-\frac{221}{3888}a^{11}+\frac{11}{243}a^{10}+\frac{935}{1296}a^{9}-\frac{55}{108}a^{8}-\frac{187}{36}a^{7}+\frac{28}{9}a^{6}+\frac{119}{6}a^{5}-\frac{28}{3}a^{4}-34a^{3}+\frac{32}{3}a^{2}+17a+1$, $\frac{1}{1679616}a^{17}-\frac{17}{279936}a^{15}+\frac{59}{23328}a^{13}-\frac{143}{2592}a^{11}+\frac{145}{216}a^{9}-\frac{967}{216}a^{7}+\frac{539}{36}a^{5}+\frac{1}{36}a^{4}-21a^{3}-\frac{2}{3}a^{2}+8a-1$, $\frac{1}{216}a^{6}-\frac{1}{6}a^{4}+\frac{1}{6}a^{3}+\frac{3}{2}a^{2}-3a+1$, $\frac{1}{10077696}a^{19}-\frac{1}{3359232}a^{18}-\frac{1}{93312}a^{17}+\frac{17}{559872}a^{16}+\frac{5}{10368}a^{15}-\frac{59}{46656}a^{14}-\frac{547}{46656}a^{13}+\frac{643}{23328}a^{12}+\frac{433}{2592}a^{11}-\frac{2597}{7776}a^{10}-\frac{1837}{1296}a^{9}+\frac{709}{324}a^{8}+\frac{251}{36}a^{7}-\frac{61}{9}a^{6}-\frac{166}{9}a^{5}+\frac{107}{18}a^{4}+\frac{43}{2}a^{3}+\frac{20}{3}a^{2}-4a-4$, $\frac{1}{559872}a^{16}-\frac{47}{279936}a^{14}+\frac{299}{46656}a^{12}+\frac{1}{7776}a^{11}-\frac{55}{432}a^{10}-\frac{5}{648}a^{9}+\frac{907}{648}a^{8}+\frac{1}{6}a^{7}-\frac{1799}{216}a^{6}-\frac{14}{9}a^{5}+\frac{869}{36}a^{4}+\frac{17}{3}a^{3}-\frac{80}{3}a^{2}-4a+6$, $\frac{1}{6}a^{2}-a+1$, $\frac{1}{10077696}a^{19}-\frac{19}{1679616}a^{17}+\frac{151}{279936}a^{15}-\frac{1}{279936}a^{14}-\frac{649}{46656}a^{13}+\frac{1}{2916}a^{12}+\frac{1625}{7776}a^{11}-\frac{17}{1296}a^{10}-\frac{263}{144}a^{9}+\frac{109}{432}a^{8}+\frac{467}{54}a^{7}-\frac{181}{72}a^{6}-\frac{217}{12}a^{5}+\frac{106}{9}a^{4}+7a^{3}-\frac{115}{6}a^{2}+8a+11$
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| Regulator: | \( 2584911672430 \) (assuming GRH) |
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| Unit signature rank: | \( 19 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 2584911672430 \cdot 2}{2\cdot\sqrt{184528125000000000000000000000000000000}}\cr\approx \mathstrut & 0.199532818164173 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15 +3 \sqrt{5}})\), 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 | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
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\(2\)
| 2.10.2.30a1.114 | $x^{20} + 2 x^{16} + 2 x^{15} + 2 x^{13} + 3 x^{12} + 4 x^{11} + 7 x^{10} + 2 x^{9} + 4 x^{8} + 4 x^{7} + 9 x^{6} + 16 x^{5} + 3 x^{4} + 8 x^{3} + 7 x^{2} + 6 x + 7$ | $2$ | $10$ | $30$ | 20T1 | not computed |
|
\(3\)
| 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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\(5\)
| 5.1.20.35a1.1 | $x^{20} + 20 x^{16} + 5$ | $20$ | $1$ | $35$ | not computed | not computed |