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Properties

Label 20.0.35066433716...0625.3
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 239^{10}$
Root discriminant $59.87$
Ramified primes $3, 5, 239$
Class number $1500$ (GRH)
Class group $[10, 150]$ (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![91872225, -14233725, -19671525, -19717650, 15232524, 3283896, 270769, -2403755, 387883, 32265, 122352, -57768, 15643, -4808, 3004, -999, 384, -123, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 40*x^18 - 123*x^17 + 384*x^16 - 999*x^15 + 3004*x^14 - 4808*x^13 + 15643*x^12 - 57768*x^11 + 122352*x^10 + 32265*x^9 + 387883*x^8 - 2403755*x^7 + 270769*x^6 + 3283896*x^5 + 15232524*x^4 - 19717650*x^3 - 19671525*x^2 - 14233725*x + 91872225)
 
gp: K = bnfinit(x^20 - 8*x^19 + 40*x^18 - 123*x^17 + 384*x^16 - 999*x^15 + 3004*x^14 - 4808*x^13 + 15643*x^12 - 57768*x^11 + 122352*x^10 + 32265*x^9 + 387883*x^8 - 2403755*x^7 + 270769*x^6 + 3283896*x^5 + 15232524*x^4 - 19717650*x^3 - 19671525*x^2 - 14233725*x + 91872225, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 40 x^{18} - 123 x^{17} + 384 x^{16} - 999 x^{15} + 3004 x^{14} - 4808 x^{13} + 15643 x^{12} - 57768 x^{11} + 122352 x^{10} + 32265 x^{9} + 387883 x^{8} - 2403755 x^{7} + 270769 x^{6} + 3283896 x^{5} + 15232524 x^{4} - 19717650 x^{3} - 19671525 x^{2} - 14233725 x + 91872225 \)

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:  \(350664337162127669847494428212890625=3^{10}\cdot 5^{10}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.87$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{45} a^{10} + \frac{1}{15} a^{6} + \frac{1}{9} a^{4} - \frac{1}{5} a^{2}$, $\frac{1}{45} a^{11} + \frac{1}{15} a^{7} + \frac{1}{9} a^{5} - \frac{1}{5} a^{3}$, $\frac{1}{135} a^{12} - \frac{1}{135} a^{10} - \frac{4}{45} a^{8} + \frac{2}{135} a^{6} - \frac{59}{135} a^{4} + \frac{1}{3} a^{3} - \frac{22}{45} a^{2} - \frac{1}{3} a$, $\frac{1}{135} a^{13} - \frac{1}{135} a^{11} + \frac{1}{45} a^{9} + \frac{2}{135} a^{7} + \frac{31}{135} a^{5} + \frac{1}{3} a^{4} + \frac{2}{5} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{405} a^{14} + \frac{1}{405} a^{13} - \frac{1}{405} a^{12} - \frac{1}{405} a^{11} - \frac{1}{135} a^{10} - \frac{4}{135} a^{9} + \frac{2}{405} a^{8} + \frac{2}{405} a^{7} - \frac{32}{405} a^{6} + \frac{121}{405} a^{5} - \frac{37}{135} a^{4} + \frac{53}{135} a^{3} - \frac{14}{45} a^{2} - \frac{1}{3} a$, $\frac{1}{4860} a^{15} - \frac{1}{810} a^{14} + \frac{4}{1215} a^{13} + \frac{1}{810} a^{12} + \frac{13}{1215} a^{11} - \frac{1}{270} a^{10} - \frac{28}{1215} a^{9} - \frac{46}{405} a^{8} - \frac{457}{4860} a^{7} - \frac{91}{810} a^{6} + \frac{1153}{2430} a^{5} - \frac{49}{135} a^{4} + \frac{613}{1620} a^{3} - \frac{11}{27} a^{2} - \frac{5}{12} a + \frac{1}{4}$, $\frac{1}{24300} a^{16} - \frac{1}{24300} a^{15} - \frac{1}{12150} a^{14} + \frac{13}{12150} a^{13} + \frac{17}{12150} a^{12} - \frac{119}{12150} a^{11} + \frac{7}{12150} a^{10} - \frac{98}{6075} a^{9} + \frac{3719}{24300} a^{8} + \frac{289}{24300} a^{7} + \frac{752}{6075} a^{6} - \frac{3337}{12150} a^{5} - \frac{1703}{8100} a^{4} - \frac{479}{1620} a^{3} + \frac{143}{540} a^{2} - \frac{1}{6} a + \frac{1}{4}$, $\frac{1}{801900} a^{17} - \frac{1}{133650} a^{16} - \frac{19}{267300} a^{15} - \frac{17}{66825} a^{14} - \frac{46}{22275} a^{13} - \frac{119}{66825} a^{12} - \frac{79}{18225} a^{11} + \frac{883}{133650} a^{10} + \frac{8333}{267300} a^{9} - \frac{8711}{133650} a^{8} - \frac{637}{9900} a^{7} + \frac{371}{12150} a^{6} - \frac{5809}{72900} a^{5} - \frac{139}{1485} a^{4} + \frac{1021}{13365} a^{3} - \frac{5761}{17820} a^{2} - \frac{19}{396} a - \frac{61}{132}$, $\frac{1}{2405700} a^{18} - \frac{1}{2405700} a^{17} + \frac{1}{200475} a^{16} + \frac{2}{66825} a^{15} - \frac{149}{400950} a^{14} - \frac{1409}{400950} a^{13} + \frac{53}{48114} a^{12} + \frac{2639}{601425} a^{11} - \frac{29}{10692} a^{10} + \frac{4487}{801900} a^{9} - \frac{28741}{400950} a^{8} - \frac{21914}{200475} a^{7} - \frac{20347}{218700} a^{6} + \frac{1167599}{2405700} a^{5} - \frac{342577}{801900} a^{4} + \frac{5891}{40095} a^{3} + \frac{1625}{10692} a^{2} - \frac{86}{297} a - \frac{37}{198}$, $\frac{1}{873640951783166045376671750376964292470500} a^{19} + \frac{37401091573918929216435695498385202}{218410237945791511344167937594241073117625} a^{18} - \frac{124235722864308836875478365221551581}{436820475891583022688335875188482146235250} a^{17} - \frac{43954877467153323269276537372172863}{14560682529719434089611195839616071541175} a^{16} - \frac{13973593381315166575156619278561292861}{145606825297194340896111958396160715411750} a^{15} + \frac{2782381452384289646265821139644732989}{3235707228826540908802487964359127009150} a^{14} + \frac{84407995621298702983098842863354107886}{218410237945791511344167937594241073117625} a^{13} + \frac{44858126363636293034241684513117702739}{218410237945791511344167937594241073117625} a^{12} + \frac{4627174999535285459519267521014760844479}{873640951783166045376671750376964292470500} a^{11} - \frac{372861474814969023618371683400668501777}{72803412648597170448055979198080357705875} a^{10} - \frac{2674547037580968481909982530944125842876}{72803412648597170448055979198080357705875} a^{9} + \frac{133734753034229148125739414971724496201}{4412328039308919421094301769580627739750} a^{8} + \frac{82090195350517646312383340290945771384681}{873640951783166045376671750376964292470500} a^{7} + \frac{43999693683889612349647409545553370307313}{436820475891583022688335875188482146235250} a^{6} + \frac{77547101845541049838257680964549897701057}{174728190356633209075334350075392858494100} a^{5} - \frac{76635027368468894671294650531926662615663}{291213650594388681792223916792321430823500} a^{4} - \frac{576985465370596998184034644241239864243}{1164854602377554727168895667169285723294} a^{3} + \frac{21707646232239549279456178470383888}{88246560786178388421886035391612554795} a^{2} + \frac{4989975974421038999684146516213862572}{21571381525510272725349919762394180061} a - \frac{110644954279481226089055126413713939}{1012740916690623132645536139079538970}$

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

Class group and class number

$C_{10}\times C_{150}$, which has order $1500$ (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:  \( 573613254.833 \) (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

\(\Q(\sqrt{3585}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-239}) \), \(\Q(\sqrt{-15}, \sqrt{-239})\), 5.5.12852225.1 x5, 10.10.592169179510490625.2, 10.0.2477695311759375.1 x5, 10.0.39477945300699375.1 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
239Data not computed