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Properties

Label 20.0.13448393894...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{14}\cdot 13^{15}$
Root discriminant $50.87$
Ramified primes $3, 5, 13$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![655664, -771760, -495304, 556458, 796807, -1239844, 432309, 271828, -242147, 5444, 51246, -14341, -2275, 3067, -423, -304, 65, 18, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + x^18 + 18*x^17 + 65*x^16 - 304*x^15 - 423*x^14 + 3067*x^13 - 2275*x^12 - 14341*x^11 + 51246*x^10 + 5444*x^9 - 242147*x^8 + 271828*x^7 + 432309*x^6 - 1239844*x^5 + 796807*x^4 + 556458*x^3 - 495304*x^2 - 771760*x + 655664)
 
gp: K = bnfinit(x^20 - x^19 + x^18 + 18*x^17 + 65*x^16 - 304*x^15 - 423*x^14 + 3067*x^13 - 2275*x^12 - 14341*x^11 + 51246*x^10 + 5444*x^9 - 242147*x^8 + 271828*x^7 + 432309*x^6 - 1239844*x^5 + 796807*x^4 + 556458*x^3 - 495304*x^2 - 771760*x + 655664, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + x^{18} + 18 x^{17} + 65 x^{16} - 304 x^{15} - 423 x^{14} + 3067 x^{13} - 2275 x^{12} - 14341 x^{11} + 51246 x^{10} + 5444 x^{9} - 242147 x^{8} + 271828 x^{7} + 432309 x^{6} - 1239844 x^{5} + 796807 x^{4} + 556458 x^{3} - 495304 x^{2} - 771760 x + 655664 \)

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:  \(13448393894735192170762921142578125=3^{16}\cdot 5^{14}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.87$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{52} a^{18} - \frac{3}{26} a^{17} + \frac{1}{26} a^{16} - \frac{3}{26} a^{14} - \frac{1}{52} a^{13} + \frac{3}{52} a^{12} - \frac{9}{52} a^{10} + \frac{1}{13} a^{9} + \frac{5}{13} a^{8} - \frac{11}{52} a^{7} - \frac{7}{26} a^{6} + \frac{23}{52} a^{5} - \frac{7}{26} a^{4} - \frac{5}{13} a^{3} + \frac{5}{26} a^{2} + \frac{4}{13} a + \frac{4}{13}$, $\frac{1}{2111871554467734448413985703897634840184984966074721580296} a^{19} + \frac{12369145118241993207565593405821801597887188812538031929}{2111871554467734448413985703897634840184984966074721580296} a^{18} + \frac{26545271664349855129724641478755469273502252392779884999}{2111871554467734448413985703897634840184984966074721580296} a^{17} + \frac{29750308303331273097781006399378562764029253572191569657}{1055935777233867224206992851948817420092492483037360790148} a^{16} - \frac{236303148345816361596901094142443099847296463214584069641}{2111871554467734448413985703897634840184984966074721580296} a^{15} + \frac{94616550770572653344650948141954165439774612821052981745}{1055935777233867224206992851948817420092492483037360790148} a^{14} - \frac{13947948903791110438762582566950226317836251584266539275}{2111871554467734448413985703897634840184984966074721580296} a^{13} - \frac{434048416592703556189185456341230404773385020142803772655}{2111871554467734448413985703897634840184984966074721580296} a^{12} + \frac{275849202573902761836532375912155213307731003255358250809}{2111871554467734448413985703897634840184984966074721580296} a^{11} - \frac{87708758542120823104365990548645585887515403396506786239}{2111871554467734448413985703897634840184984966074721580296} a^{10} + \frac{54258314895786035686905262064671074118595109823118693645}{1055935777233867224206992851948817420092492483037360790148} a^{9} - \frac{370607139875179403175746872547685647342682536616776523135}{1055935777233867224206992851948817420092492483037360790148} a^{8} + \frac{127108900504955081140206100783458789578394426233012040451}{2111871554467734448413985703897634840184984966074721580296} a^{7} - \frac{87240967706642311435962351501331797895386909011701846021}{1055935777233867224206992851948817420092492483037360790148} a^{6} - \frac{445890117139942172526290944528776563454384050241474432037}{2111871554467734448413985703897634840184984966074721580296} a^{5} - \frac{136384043654470734189559695771924622301800456159432933343}{527967888616933612103496425974408710046246241518680395074} a^{4} - \frac{511307314432336501689796523327799448138897428405739099231}{2111871554467734448413985703897634840184984966074721580296} a^{3} + \frac{324157541308906932863724625228615069777284548228916588911}{1055935777233867224206992851948817420092492483037360790148} a^{2} - \frac{21067285402264245904753241834608888326844226979837787531}{527967888616933612103496425974408710046246241518680395074} a + \frac{114502799515733012244014674650604259740427255474425125427}{263983944308466806051748212987204355023123120759340197537}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( 63030233.332374446 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.54925.1, 5.1.1711125.1, 10.2.38063333953125.2

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $20$

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$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
13Data not computed