Normalized defining polynomial
\( x^{30} + 29 x^{28} + 378 x^{26} + 2925 x^{24} + 14950 x^{22} + 53130 x^{20} + 134596 x^{18} + 245157 x^{16} + \cdots + 1 \)
Invariants
| Degree: | $30$ |
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| Signature: | $[0, 15]$ |
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| Discriminant: |
\(-615215540441622698713738389172402189599059721846784\)
\(\medspace = -\,2^{30}\cdot 31^{28}\)
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| Root discriminant: | \(49.31\) |
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| Galois root discriminant: | $2\cdot 31^{14/15}\approx 49.31369861109494$ | ||
| Ramified primes: |
\(2\), \(31\)
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| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{30}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(124=2^{2}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{124}(1,·)$, $\chi_{124}(67,·)$, $\chi_{124}(97,·)$, $\chi_{124}(5,·)$, $\chi_{124}(7,·)$, $\chi_{124}(9,·)$, $\chi_{124}(111,·)$, $\chi_{124}(109,·)$, $\chi_{124}(81,·)$, $\chi_{124}(19,·)$, $\chi_{124}(87,·)$, $\chi_{124}(25,·)$, $\chi_{124}(103,·)$, $\chi_{124}(69,·)$, $\chi_{124}(107,·)$, $\chi_{124}(33,·)$, $\chi_{124}(35,·)$, $\chi_{124}(113,·)$, $\chi_{124}(101,·)$, $\chi_{124}(39,·)$, $\chi_{124}(41,·)$, $\chi_{124}(95,·)$, $\chi_{124}(71,·)$, $\chi_{124}(45,·)$, $\chi_{124}(47,·)$, $\chi_{124}(49,·)$, $\chi_{124}(51,·)$, $\chi_{124}(121,·)$, $\chi_{124}(59,·)$, $\chi_{124}(63,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{16384}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2542}$, which has order $5084$ (assuming GRH) |
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| Narrow class group: | $C_{2542}\times C_{2}$, which has order $5084$ (assuming GRH) |
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| Relative class number: | $5084$ (assuming GRH) |
Unit group
| Rank: | $14$ |
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| Torsion generator: |
\( a^{29} + 28 a^{27} + 351 a^{25} + 2600 a^{23} + 12650 a^{21} + 42504 a^{19} + 100947 a^{17} + 170544 a^{15} + 203490 a^{13} + 167960 a^{11} + 92378 a^{9} + 31824 a^{7} + 6188 a^{5} + 560 a^{3} + 15 a \)
(order $4$)
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| Fundamental units: |
$a^{29}+28a^{27}+350a^{25}+2576a^{23}+12398a^{21}+40984a^{19}+95132a^{17}+155840a^{15}+178634a^{13}+140152a^{11}+72412a^{9}+23136a^{7}+4116a^{5}+336a^{3}+8a$, $a$, $a^{15}+15a^{13}+90a^{11}+275a^{9}+450a^{7}+378a^{5}+140a^{3}+15a$, $a^{27}+27a^{25}+324a^{23}+2277a^{21}+10395a^{19}+32319a^{17}+69768a^{15}+104652a^{13}+107406a^{11}+72930a^{9}+30888a^{7}+7371a^{5}+819a^{3}+27a$, $a^{25}+25a^{23}+275a^{21}+1750a^{19}+7125a^{17}+19380a^{15}+35700a^{13}+44200a^{11}+35750a^{9}+17875a^{7}+5005a^{5}+650a^{3}+25a$, $a^{4}+4a^{2}+3$, $a^{19}+19a^{17}+152a^{15}+665a^{13}+1729a^{11}+2717a^{9}+2508a^{7}+1254a^{5}+285a^{3}+19a$, $a^{21}+20a^{19}+170a^{17}+800a^{15}+2275a^{13}+4004a^{11}+4290a^{9}+2640a^{7}+825a^{5}+100a^{3}+3a$, $a^{29}+28a^{27}+351a^{25}+2600a^{23}+12649a^{21}+42483a^{19}+100758a^{17}+169592a^{15}+200550a^{13}+162228a^{11}+85382a^{9}+26720a^{7}+4186a^{5}+230a^{3}+4a$, $a^{28}+27a^{26}+324a^{24}+2277a^{22}+10396a^{20}+32338a^{18}+69920a^{16}+105316a^{14}+109122a^{12}+75581a^{10}+33230a^{8}+8407a^{6}+958a^{4}+2a^{2}-4$, $a^{22}+22a^{20}+209a^{18}+1122a^{16}+3740a^{14}+8008a^{12}+11011a^{10}+9438a^{8}+4719a^{6}+1210a^{4}+121a^{2}+1$, $a^{18}+18a^{16}+135a^{14}+546a^{12}+1287a^{10}+1782a^{8}+1386a^{6}+540a^{4}+81a^{2}+1$, $a^{3}+3a$, $a^{29}+29a^{27}+378a^{25}+2925a^{23}+14950a^{21}+53130a^{19}+134596a^{17}+245156a^{15}+319755a^{13}+293839a^{11}+184470a^{9}+75087a^{7}+18102a^{5}+2170a^{3}+84a$
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| Regulator: | \( 4316173757.895952 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 4316173757.895952 \cdot 5084}{4\cdot\sqrt{615215540441622698713738389172402189599059721846784}}\cr\approx \mathstrut & 0.207696341766930 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 3.3.961.1, 5.5.923521.1, 6.0.59105344.1, 10.0.873360422339584.1, \(\Q(\zeta_{31})^+\) |
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 | $30$ | ${\href{/padicField/5.3.0.1}{3} }^{10}$ | $30$ | $30$ | $15^{2}$ | $15^{2}$ | $30$ | ${\href{/padicField/23.10.0.1}{10} }^{3}$ | ${\href{/padicField/29.5.0.1}{5} }^{6}$ | R | ${\href{/padicField/37.3.0.1}{3} }^{10}$ | $15^{2}$ | $30$ | ${\href{/padicField/47.10.0.1}{10} }^{3}$ | $15^{2}$ | $30$ |
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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\(2\)
| 2.5.2.10a1.1 | $x^{10} + 2 x^{7} + 4 x^{5} + x^{4} + 4 x^{2} + 5$ | $2$ | $5$ | $10$ | $C_{10}$ | $$[2]^{5}$$ |
| 2.5.2.10a1.1 | $x^{10} + 2 x^{7} + 4 x^{5} + x^{4} + 4 x^{2} + 5$ | $2$ | $5$ | $10$ | $C_{10}$ | $$[2]^{5}$$ | |
| 2.5.2.10a1.1 | $x^{10} + 2 x^{7} + 4 x^{5} + x^{4} + 4 x^{2} + 5$ | $2$ | $5$ | $10$ | $C_{10}$ | $$[2]^{5}$$ | |
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\(31\)
| Deg $30$ | $15$ | $2$ | $28$ |