Normalized defining polynomial
\( x^{30} - 30 x^{28} - x^{27} + 405 x^{26} + 27 x^{25} - 3250 x^{24} - 324 x^{23} + 17250 x^{22} + \cdots + 1 \)
Invariants
Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[30, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(38731022422755868071217069332926190761458900976553\) \(\medspace = 3^{45}\cdot 11^{27}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(44.97\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $3^{3/2}11^{9/10}\approx 44.971285159244935$ | ||
Ramified primes: | \(3\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{33}) \) | ||
$\card{ \Gal(K/\Q) }$: | $30$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(99=3^{2}\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{99}(64,·)$, $\chi_{99}(1,·)$, $\chi_{99}(2,·)$, $\chi_{99}(67,·)$, $\chi_{99}(4,·)$, $\chi_{99}(68,·)$, $\chi_{99}(65,·)$, $\chi_{99}(8,·)$, $\chi_{99}(74,·)$, $\chi_{99}(98,·)$, $\chi_{99}(16,·)$, $\chi_{99}(17,·)$, $\chi_{99}(82,·)$, $\chi_{99}(83,·)$, $\chi_{99}(25,·)$, $\chi_{99}(91,·)$, $\chi_{99}(29,·)$, $\chi_{99}(31,·)$, $\chi_{99}(32,·)$, $\chi_{99}(97,·)$, $\chi_{99}(34,·)$, $\chi_{99}(35,·)$, $\chi_{99}(37,·)$, $\chi_{99}(41,·)$, $\chi_{99}(49,·)$, $\chi_{99}(50,·)$, $\chi_{99}(58,·)$, $\chi_{99}(95,·)$, $\chi_{99}(70,·)$, $\chi_{99}(62,·)$$\rbrace$ | ||
This is not a CM field. |
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
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $29$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+4719a^{6}-1210a^{4}+121a^{2}-2$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a-1$, $a^{27}-27a^{25}+324a^{23}-2278a^{21}+10416a^{19}+a^{18}-32508a^{17}-18a^{16}+70720a^{15}+135a^{14}-107592a^{13}-547a^{12}+113139a^{11}+1299a^{10}-79936a^{9}-1836a^{8}+36027a^{7}+1498a^{6}-9423a^{5}-645a^{4}+1174a^{3}+117a^{2}-39a-3$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}+a^{9}-1782a^{8}-9a^{7}+1386a^{6}+27a^{5}-540a^{4}-30a^{3}+81a^{2}+9a-1$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a+1$, $a^{27}-27a^{25}+324a^{23}-2278a^{21}+10416a^{19}+a^{18}-32508a^{17}-18a^{16}+70720a^{15}+135a^{14}-107592a^{13}-547a^{12}+113139a^{11}+1299a^{10}-79936a^{9}-1836a^{8}+36027a^{7}+1498a^{6}-9423a^{5}-645a^{4}+1173a^{3}+117a^{2}-36a-4$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a-1$, $a^{6}-6a^{4}+9a^{2}-3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+1$, $a^{24}-24a^{22}-a^{21}+252a^{20}+21a^{19}-1520a^{18}-189a^{17}+5814a^{16}+952a^{15}-14688a^{14}-2940a^{13}+24751a^{12}+5733a^{11}-27444a^{10}-7006a^{9}+19251a^{8}+5139a^{7}-7896a^{6}-2052a^{5}+1611a^{4}+355a^{3}-108a^{2}-12a$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+a^{7}+13013a^{6}-7a^{5}-2366a^{4}+14a^{3}+169a^{2}-7a-3$, $a^{7}-7a^{5}+14a^{3}-7a+1$, $a^{27}-28a^{25}+349a^{23}-2552a^{21}+12145a^{19}-39445a^{17}+89165a^{15}-140470a^{13}+152034a^{11}-109539a^{9}-a^{8}+49685a^{7}+8a^{6}-12831a^{5}-20a^{4}+1526a^{3}+16a^{2}-42a-2$, $a^{2}-1$, $a^{4}-4a^{2}+3$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a+1$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+3$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3741a^{14}-8022a^{12}+a^{11}+11088a^{10}-11a^{9}-9648a^{8}+44a^{7}+5013a^{6}-77a^{5}-1406a^{4}+55a^{3}+170a^{2}-11a-4$, $a^{5}-5a^{3}+5a$, $a^{5}-5a^{3}+5a+1$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}+a^{10}-16445a^{9}-10a^{8}+9867a^{7}+35a^{6}-3289a^{5}-50a^{4}+506a^{3}+25a^{2}-23a-3$, $a^{7}-7a^{5}+14a^{3}-7a$, $a^{28}-28a^{26}+350a^{24}-2577a^{22}+12419a^{20}-41173a^{18}+a^{17}+96084a^{16}-17a^{15}-158780a^{14}+119a^{13}+184366a^{12}-443a^{11}-147146a^{10}+946a^{9}+77496a^{8}-1167a^{7}-25068a^{6}+798a^{5}+4345a^{4}-273a^{3}-292a^{2}+36a+2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+a^{13}+2275a^{12}-13a^{11}-4004a^{10}+65a^{9}+4290a^{8}-156a^{7}-2640a^{6}+182a^{5}+825a^{4}-91a^{3}-100a^{2}+13a+1$, $a^{26}-26a^{24}+299a^{22}-2003a^{20}+8666a^{18}-25382a^{16}+51323a^{14}-71774a^{12}+68510a^{10}-43315a^{8}+17178a^{6}-3867a^{4}+399a^{2}-9$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}+a^{4}-1015a^{3}-4a^{2}+29a+2$, $a^{28}-29a^{26}-a^{25}+376a^{24}+26a^{23}-2874a^{22}-298a^{21}+14376a^{20}+1980a^{19}-49380a^{18}-8436a^{17}+118864a^{16}+24072a^{15}-200888a^{14}-46648a^{13}+235027a^{12}+60944a^{11}-184327a^{10}-52196a^{9}+91519a^{8}+27750a^{7}-26030a^{6}-8313a^{5}+3496a^{4}+1166a^{3}-129a^{2}-42a$, $a^{28}-27a^{26}+a^{25}+324a^{24}-25a^{23}-2277a^{22}+275a^{21}+10395a^{20}-1749a^{19}-32319a^{18}+7107a^{17}+69769a^{16}-19245a^{15}-104668a^{14}+35154a^{13}+107510a^{12}-42913a^{11}-73281a^{10}+33968a^{9}+31539a^{8}-16488a^{7}-8016a^{6}+4458a^{5}+1125a^{4}-555a^{3}-82a^{2}+17a+2$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 1458908175124897.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{30}\cdot(2\pi)^{0}\cdot 1458908175124897.5 \cdot 1}{2\cdot\sqrt{38731022422755868071217069332926190761458900976553}}\cr\approx \mathstrut & 0.125854384871860 \end{aligned}\] (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
\(\Q(\sqrt{33}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), 6.6.26198073.1, \(\Q(\zeta_{33})^+\), 15.15.10943023107606534329121.1 |
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 | $15^{2}$ | R | $30$ | $30$ | R | $30$ | ${\href{/padicField/17.5.0.1}{5} }^{6}$ | ${\href{/padicField/19.10.0.1}{10} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{5}$ | $15^{2}$ | $15^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{6}$ | $15^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{5}$ | $30$ | ${\href{/padicField/53.10.0.1}{10} }^{3}$ | $30$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $30$ | $6$ | $5$ | $45$ | |||
\(11\) | Deg $30$ | $10$ | $3$ | $27$ |