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Properties

Label 20.0.58673144680...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{35}\cdot 17^{10}$
Root discriminant $68.93$
Ramified primes $5, 17$
Class number $58322$ (GRH)
Class group $[11, 5302]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![161532401, -123793680, 98054400, -82660420, 89987200, -40331421, 49539760, -11635200, 18020840, -1777600, 4111321, -145440, 582400, -6060, 51200, -101, 2720, 0, 80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 80*x^18 + 2720*x^16 - 101*x^15 + 51200*x^14 - 6060*x^13 + 582400*x^12 - 145440*x^11 + 4111321*x^10 - 1777600*x^9 + 18020840*x^8 - 11635200*x^7 + 49539760*x^6 - 40331421*x^5 + 89987200*x^4 - 82660420*x^3 + 98054400*x^2 - 123793680*x + 161532401)
 
gp: K = bnfinit(x^20 + 80*x^18 + 2720*x^16 - 101*x^15 + 51200*x^14 - 6060*x^13 + 582400*x^12 - 145440*x^11 + 4111321*x^10 - 1777600*x^9 + 18020840*x^8 - 11635200*x^7 + 49539760*x^6 - 40331421*x^5 + 89987200*x^4 - 82660420*x^3 + 98054400*x^2 - 123793680*x + 161532401, 1)
 

Normalized defining polynomial

\( x^{20} + 80 x^{18} + 2720 x^{16} - 101 x^{15} + 51200 x^{14} - 6060 x^{13} + 582400 x^{12} - 145440 x^{11} + 4111321 x^{10} - 1777600 x^{9} + 18020840 x^{8} - 11635200 x^{7} + 49539760 x^{6} - 40331421 x^{5} + 89987200 x^{4} - 82660420 x^{3} + 98054400 x^{2} - 123793680 x + 161532401 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5867314468047698028385639190673828125=5^{35}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.93$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(425=5^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{425}(256,·)$, $\chi_{425}(1,·)$, $\chi_{425}(322,·)$, $\chi_{425}(67,·)$, $\chi_{425}(324,·)$, $\chi_{425}(69,·)$, $\chi_{425}(203,·)$, $\chi_{425}(341,·)$, $\chi_{425}(86,·)$, $\chi_{425}(407,·)$, $\chi_{425}(152,·)$, $\chi_{425}(409,·)$, $\chi_{425}(154,·)$, $\chi_{425}(288,·)$, $\chi_{425}(33,·)$, $\chi_{425}(171,·)$, $\chi_{425}(237,·)$, $\chi_{425}(239,·)$, $\chi_{425}(373,·)$, $\chi_{425}(118,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{29} a^{10} + \frac{11}{29} a^{8} + \frac{9}{29} a^{6} - \frac{7}{29} a^{5} + \frac{10}{29} a^{4} + \frac{5}{29} a^{3} - \frac{9}{29} a^{2} - \frac{9}{29} a + \frac{9}{29}$, $\frac{1}{29} a^{11} + \frac{11}{29} a^{9} + \frac{9}{29} a^{7} - \frac{7}{29} a^{6} + \frac{10}{29} a^{5} + \frac{5}{29} a^{4} - \frac{9}{29} a^{3} - \frac{9}{29} a^{2} + \frac{9}{29} a$, $\frac{1}{29} a^{12} + \frac{4}{29} a^{8} - \frac{7}{29} a^{7} - \frac{2}{29} a^{6} - \frac{5}{29} a^{5} - \frac{3}{29} a^{4} - \frac{6}{29} a^{3} - \frac{8}{29} a^{2} + \frac{12}{29} a - \frac{12}{29}$, $\frac{1}{3956947021} a^{13} - \frac{62734977}{3956947021} a^{12} + \frac{52}{3956947021} a^{11} - \frac{9457018}{3956947021} a^{10} + \frac{1040}{3956947021} a^{9} - \frac{33779875}{3956947021} a^{8} - \frac{1091561608}{3956947021} a^{7} + \frac{861859462}{3956947021} a^{6} - \frac{136399857}{3956947021} a^{5} - \frac{929643712}{3956947021} a^{4} - \frac{1637264204}{3956947021} a^{3} - \frac{986453170}{3956947021} a^{2} - \frac{1500857691}{3956947021} a - \frac{204051099}{3956947021}$, $\frac{1}{3956947021} a^{14} + \frac{56}{3956947021} a^{12} - \frac{21952990}{3956947021} a^{11} + \frac{1232}{3956947021} a^{10} + \frac{1490104522}{3956947021} a^{9} + \frac{13440}{3956947021} a^{8} + \frac{1737347614}{3956947021} a^{7} - \frac{1637282124}{3956947021} a^{6} - \frac{800979357}{3956947021} a^{5} - \frac{1091370888}{3956947021} a^{4} - \frac{1547803154}{3956947021} a^{3} - \frac{409138643}{3956947021} a^{2} + \frac{1872736514}{3956947021} a + \frac{32768}{3956947021}$, $\frac{1}{3956947021} a^{15} - \frac{56401952}{3956947021} a^{12} - \frac{1680}{3956947021} a^{11} - \frac{26999205}{3956947021} a^{10} - \frac{44800}{3956947021} a^{9} - \frac{1419497999}{3956947021} a^{8} + \frac{1227534201}{3956947021} a^{7} - \frac{1035959181}{3956947021} a^{6} - \frac{957533591}{3956947021} a^{5} + \frac{1118630180}{3956947021} a^{4} - \frac{550803396}{3956947021} a^{3} + \frac{1034623495}{3956947021} a^{2} + \frac{542836676}{3956947021} a - \frac{34640172}{3956947021}$, $\frac{1}{3956947021} a^{16} - \frac{1920}{3956947021} a^{12} + \frac{40526870}{3956947021} a^{11} - \frac{56320}{3956947021} a^{10} + \frac{1977720236}{3956947021} a^{9} - \frac{691200}{3956947021} a^{8} - \frac{78240293}{3956947021} a^{7} - \frac{413468115}{3956947021} a^{6} - \frac{1143373160}{3956947021} a^{5} - \frac{284361698}{3956947021} a^{4} - \frac{1579132899}{3956947021} a^{3} - \frac{1103368072}{3956947021} a^{2} + \frac{1434621924}{3956947021} a - \frac{1966080}{3956947021}$, $\frac{1}{3956947021} a^{17} - \frac{64860952}{3956947021} a^{12} + \frac{43520}{3956947021} a^{11} + \frac{1978383}{136446449} a^{10} + \frac{1305600}{3956947021} a^{9} + \frac{1513820370}{3956947021} a^{8} + \frac{47569138}{136446449} a^{7} + \frac{32278449}{3956947021} a^{6} - \frac{194905058}{3956947021} a^{5} + \frac{1226314859}{3956947021} a^{4} + \frac{43403048}{136446449} a^{3} + \frac{1867033440}{3956947021} a^{2} - \frac{854855063}{3956947021} a - \frac{176801450}{3956947021}$, $\frac{1}{3956947021} a^{18} + \frac{52224}{3956947021} a^{12} + \frac{18981386}{3956947021} a^{11} + \frac{1723392}{3956947021} a^{10} - \frac{209139193}{3956947021} a^{9} + \frac{22560768}{3956947021} a^{8} + \frac{711408508}{3956947021} a^{7} - \frac{132514786}{3956947021} a^{6} + \frac{529605746}{3956947021} a^{5} - \frac{1099830619}{3956947021} a^{4} - \frac{481509929}{3956947021} a^{3} - \frac{1625562399}{3956947021} a^{2} - \frac{1694257381}{3956947021} a + \frac{71303168}{3956947021}$, $\frac{1}{3956947021} a^{19} - \frac{61713154}{3956947021} a^{12} - \frac{992256}{3956947021} a^{11} + \frac{10916357}{3956947021} a^{10} - \frac{31752192}{3956947021} a^{9} - \frac{1331235348}{3956947021} a^{8} + \frac{1802116880}{3956947021} a^{7} + \frac{1735658378}{3956947021} a^{6} + \frac{969681590}{3956947021} a^{5} - \frac{870884723}{3956947021} a^{4} + \frac{1967676223}{3956947021} a^{3} - \frac{1339405945}{3956947021} a^{2} + \frac{974533739}{3956947021} a + \frac{988498868}{3956947021}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{11}\times C_{5302}$, which has order $58322$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 161406.837641 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.36125.1, 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 $20$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
17Data not computed