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Properties

Label 20.0.30368554900...1249.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 401^{10}$
Root discriminant $52.98$
Ramified primes $7, 401$
Class number $2106$ (GRH)
Class group $[3, 3, 3, 78]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8429607, -25841940, 40346065, -41598580, 31602475, -18484780, 8187004, -2617976, 575097, -87050, 37971, -26648, 16756, -9518, 4193, -1636, 507, -136, 35, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 35*x^18 - 136*x^17 + 507*x^16 - 1636*x^15 + 4193*x^14 - 9518*x^13 + 16756*x^12 - 26648*x^11 + 37971*x^10 - 87050*x^9 + 575097*x^8 - 2617976*x^7 + 8187004*x^6 - 18484780*x^5 + 31602475*x^4 - 41598580*x^3 + 40346065*x^2 - 25841940*x + 8429607)
 
gp: K = bnfinit(x^20 - 6*x^19 + 35*x^18 - 136*x^17 + 507*x^16 - 1636*x^15 + 4193*x^14 - 9518*x^13 + 16756*x^12 - 26648*x^11 + 37971*x^10 - 87050*x^9 + 575097*x^8 - 2617976*x^7 + 8187004*x^6 - 18484780*x^5 + 31602475*x^4 - 41598580*x^3 + 40346065*x^2 - 25841940*x + 8429607, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 35 x^{18} - 136 x^{17} + 507 x^{16} - 1636 x^{15} + 4193 x^{14} - 9518 x^{13} + 16756 x^{12} - 26648 x^{11} + 37971 x^{10} - 87050 x^{9} + 575097 x^{8} - 2617976 x^{7} + 8187004 x^{6} - 18484780 x^{5} + 31602475 x^{4} - 41598580 x^{3} + 40346065 x^{2} - 25841940 x + 8429607 \)

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:  \(30368554900942376860763008296271249=7^{10}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.98$
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:  $7, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{13} + \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{2} a^{8} - \frac{1}{18} a^{6} - \frac{1}{18} a^{5} - \frac{1}{2} a^{4} + \frac{4}{9} a^{3} + \frac{1}{18} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{13} + \frac{1}{18} a^{12} + \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} + \frac{4}{9} a^{7} - \frac{1}{2} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{18} a^{2} + \frac{1}{6} a$, $\frac{1}{54} a^{16} - \frac{1}{54} a^{15} + \frac{1}{18} a^{13} + \frac{2}{27} a^{11} + \frac{1}{54} a^{10} - \frac{1}{9} a^{9} + \frac{13}{27} a^{8} - \frac{4}{27} a^{7} - \frac{1}{2} a^{6} + \frac{5}{18} a^{5} + \frac{1}{3} a^{4} - \frac{11}{27} a^{3} - \frac{1}{54} a^{2} + \frac{5}{18} a - \frac{1}{3}$, $\frac{1}{2106} a^{17} + \frac{5}{1053} a^{16} - \frac{11}{2106} a^{15} + \frac{5}{702} a^{14} + \frac{5}{117} a^{13} + \frac{29}{1053} a^{12} + \frac{1}{54} a^{11} + \frac{1}{1053} a^{10} - \frac{110}{1053} a^{9} - \frac{509}{1053} a^{8} - \frac{206}{1053} a^{7} - \frac{53}{702} a^{6} - \frac{103}{234} a^{5} + \frac{797}{2106} a^{4} + \frac{5}{54} a^{3} + \frac{20}{81} a^{2} + \frac{50}{351} a - \frac{73}{234}$, $\frac{1}{8500433058} a^{18} - \frac{239132}{4250216529} a^{17} + \frac{35360027}{8500433058} a^{16} - \frac{50861632}{4250216529} a^{15} - \frac{58603249}{2833477686} a^{14} - \frac{23779465}{4250216529} a^{13} + \frac{550964143}{8500433058} a^{12} - \frac{169380899}{4250216529} a^{11} + \frac{185813267}{4250216529} a^{10} + \frac{1372494115}{8500433058} a^{9} + \frac{1377978193}{8500433058} a^{8} + \frac{326268706}{4250216529} a^{7} + \frac{434274055}{2833477686} a^{6} - \frac{1268020669}{8500433058} a^{5} + \frac{35955623}{274207518} a^{4} + \frac{120159011}{326939733} a^{3} - \frac{323738902}{4250216529} a^{2} - \frac{1188680189}{2833477686} a + \frac{73321553}{944492562}$, $\frac{1}{25864459598761020663095909911885967214486820314} a^{19} + \frac{1399103490615305805717818111710960711}{25864459598761020663095909911885967214486820314} a^{18} + \frac{1351700653536762383532510362995144548937}{6045923234866998752476837286555859563928663} a^{17} + \frac{44914927619392587898968654982667091403459307}{25864459598761020663095909911885967214486820314} a^{16} - \frac{468424808156448945450320281935973103725944421}{25864459598761020663095909911885967214486820314} a^{15} - \frac{215924460173563510029438501439593821025157399}{12932229799380510331547954955942983607243410157} a^{14} + \frac{159663953728647897191919463417738464655782851}{8621486532920340221031969970628655738162273438} a^{13} - \frac{923061502106239139322791939593980090368659375}{12932229799380510331547954955942983607243410157} a^{12} + \frac{104698057257811470656688891286073714219692615}{12932229799380510331547954955942983607243410157} a^{11} - \frac{4903479572361432572562967101597412818003113}{110531878627183848987589358597803278694388121} a^{10} - \frac{25050302155108065046209889568003684068970885}{159657158017043337426517962419049180336338397} a^{9} + \frac{3667025665304238911517290631171259307193601}{1989573815289309281776608454760459016498986178} a^{8} - \frac{504180741667552193045727390621877972156178857}{1124541721685261767960691735299389878890731318} a^{7} - \frac{6232305806039574457512890046823760871057429247}{25864459598761020663095909911885967214486820314} a^{6} - \frac{148420035102606818176873783331998508042599999}{4310743266460170110515984985314327869081136719} a^{5} + \frac{27606801034054414788544607676714192711200257}{994786907644654640888304227380229508249493089} a^{4} + \frac{1047652576883231271985493375900134179786522575}{2873828844306780073677323323542885246054091146} a^{3} - \frac{5382532883104667406447183040077429158055099757}{25864459598761020663095909911885967214486820314} a^{2} + \frac{1205034441990325751745123996037032966472104773}{8621486532920340221031969970628655738162273438} a - \frac{49883868230993143466650301138924686921140256}{110531878627183848987589358597803278694388121}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{78}$, which has order $2106$ (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:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{401}) \), \(\Q(\sqrt{-2807}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-7}, \sqrt{401})\), 5.5.160801.1 x5, 10.10.10368641602001.1, 10.0.174265759404830807.1 x5, 10.0.434577953628007.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$7$7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
401Data not computed