Normalized defining polynomial
\( x^{29} - x^{28} - 28 x^{27} + 27 x^{26} + 351 x^{25} - 325 x^{24} - 2600 x^{23} + 2300 x^{22} + 12650 x^{21} - 10626 x^{20} - 42504 x^{19} + 33649 x^{18} + 100947 x^{17} - 74613 x^{16} + \cdots - 1 \)
Invariants
Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[29, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(38358032782038398419973086399760468678777161743121\) \(\medspace = 59^{28}\) | 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: | \(51.26\) | 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: | $59^{28/29}\approx 51.261126887866126$ | ||
Ramified primes: | \(59\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $29$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(59\) | ||
Dirichlet character group: | $\lbrace$$\chi_{59}(1,·)$, $\chi_{59}(3,·)$, $\chi_{59}(4,·)$, $\chi_{59}(5,·)$, $\chi_{59}(7,·)$, $\chi_{59}(9,·)$, $\chi_{59}(12,·)$, $\chi_{59}(15,·)$, $\chi_{59}(16,·)$, $\chi_{59}(17,·)$, $\chi_{59}(19,·)$, $\chi_{59}(20,·)$, $\chi_{59}(21,·)$, $\chi_{59}(22,·)$, $\chi_{59}(25,·)$, $\chi_{59}(26,·)$, $\chi_{59}(27,·)$, $\chi_{59}(28,·)$, $\chi_{59}(29,·)$, $\chi_{59}(35,·)$, $\chi_{59}(36,·)$, $\chi_{59}(41,·)$, $\chi_{59}(45,·)$, $\chi_{59}(46,·)$, $\chi_{59}(48,·)$, $\chi_{59}(49,·)$, $\chi_{59}(51,·)$, $\chi_{59}(53,·)$, $\chi_{59}(57,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $28$ | 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: | $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a$, $a^{5}-5a^{3}+5a$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-1$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{25}-25a^{23}+275a^{21}-a^{20}-1750a^{19}+20a^{18}+7125a^{17}-170a^{16}-19379a^{15}+800a^{14}+35685a^{13}-2275a^{12}-44110a^{11}+4003a^{10}+35475a^{9}-4280a^{8}-17425a^{7}+2605a^{6}+4628a^{5}-775a^{4}-515a^{3}+75a^{2}+15a-1$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12398a^{20}-40984a^{18}+95132a^{16}-155840a^{14}+178634a^{12}-140152a^{10}+72412a^{8}-23136a^{6}+4116a^{4}-336a^{2}+8$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19306a^{8}-8016a^{6}+1736a^{4}-160a^{2}+4$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a$, $a^{28}-27a^{26}+325a^{24}-2300a^{22}+10626a^{20}-33649a^{18}+74613a^{16}-116280a^{14}+125970a^{12}-92378a^{10}+43758a^{8}-12376a^{6}+1820a^{4}-105a^{2}+1$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-1$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{5}-5a^{3}+5a-1$, $a^{3}-3a-1$, $a^{28}-28a^{26}-a^{25}+350a^{24}+25a^{23}-2575a^{22}-275a^{21}+12375a^{20}+1750a^{19}-40755a^{18}-7125a^{17}+93840a^{16}+19380a^{15}-151300a^{14}-35700a^{13}+168350a^{12}+44200a^{11}-125125a^{10}-35751a^{9}+58630a^{8}+17884a^{7}-15664a^{6}-5032a^{5}+1969a^{4}+679a^{3}-66a^{2}-31a-1$, $a^{7}-7a^{5}+14a^{3}-7a$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}-a^{19}+8645a^{18}+19a^{17}-25194a^{16}-152a^{15}+50388a^{14}+665a^{13}-68952a^{12}-1729a^{11}+63206a^{10}+2717a^{9}-37180a^{8}-2509a^{7}+13013a^{6}+1261a^{5}-2366a^{4}-299a^{3}+169a^{2}+26a-1$, $a^{21}-21a^{19}+189a^{17}-952a^{15}-a^{14}+2940a^{13}+14a^{12}-5733a^{11}-77a^{10}+7007a^{9}+210a^{8}-5147a^{7}-294a^{6}+2072a^{5}+196a^{4}-371a^{3}-49a^{2}+14a+1$, $a^{28}-a^{27}-27a^{26}+27a^{25}+324a^{24}-325a^{23}-2276a^{22}+2299a^{21}+10373a^{20}-10604a^{19}-32109a^{18}+33440a^{17}+68628a^{16}-73491a^{15}-100777a^{14}+112540a^{13}+98853a^{12}-117962a^{11}-60644a^{10}+81368a^{9}+19723a^{8}-34330a^{7}-1386a^{6}+7692a^{5}-806a^{4}-660a^{3}+124a^{2}+9a-2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{6}-6a^{4}+9a^{2}-1$ (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: | \( 2275980944744796.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{29}\cdot(2\pi)^{0}\cdot 2275980944744796.5 \cdot 1}{2\cdot\sqrt{38358032782038398419973086399760468678777161743121}}\cr\approx \mathstrut & 0.0986461947872962 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 29 |
The 29 conjugacy class representatives for $C_{29}$ |
Character table for $C_{29}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | $29$ | R |
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 |
---|---|---|---|---|---|---|---|
\(59\) | Deg $29$ | $29$ | $1$ | $28$ |