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Properties

Label 20.0.51315690279...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{16}\cdot 5^{23}$
Root discriminant $15.33$
Ramified primes $3, 5$
Class number $1$
Class group Trivial
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, -25, -30, -20, -20, -1, 60, -155, 420, -560, 711, -780, 770, -645, 460, -281, 140, -60, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 20*x^18 - 60*x^17 + 140*x^16 - 281*x^15 + 460*x^14 - 645*x^13 + 770*x^12 - 780*x^11 + 711*x^10 - 560*x^9 + 420*x^8 - 155*x^7 + 60*x^6 - x^5 - 20*x^4 - 20*x^3 - 30*x^2 - 25*x + 41)
 
gp: K = bnfinit(x^20 - 5*x^19 + 20*x^18 - 60*x^17 + 140*x^16 - 281*x^15 + 460*x^14 - 645*x^13 + 770*x^12 - 780*x^11 + 711*x^10 - 560*x^9 + 420*x^8 - 155*x^7 + 60*x^6 - x^5 - 20*x^4 - 20*x^3 - 30*x^2 - 25*x + 41, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 20 x^{18} - 60 x^{17} + 140 x^{16} - 281 x^{15} + 460 x^{14} - 645 x^{13} + 770 x^{12} - 780 x^{11} + 711 x^{10} - 560 x^{9} + 420 x^{8} - 155 x^{7} + 60 x^{6} - x^{5} - 20 x^{4} - 20 x^{3} - 30 x^{2} - 25 x + 41 \)

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:  \(513156902790069580078125=3^{16}\cdot 5^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.33$
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:  $3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{82} a^{16} - \frac{8}{41} a^{15} + \frac{9}{41} a^{14} - \frac{1}{82} a^{13} - \frac{5}{82} a^{12} - \frac{4}{41} a^{11} + \frac{9}{82} a^{10} - \frac{27}{82} a^{9} - \frac{12}{41} a^{8} + \frac{5}{82} a^{7} - \frac{23}{82} a^{6} - \frac{9}{41} a^{5} - \frac{13}{82} a^{4} - \frac{23}{82} a^{3} + \frac{11}{41} a^{2} - \frac{10}{41} a - \frac{1}{2}$, $\frac{1}{82} a^{17} + \frac{4}{41} a^{15} + \frac{10}{41} a^{13} - \frac{3}{41} a^{12} + \frac{2}{41} a^{11} - \frac{3}{41} a^{10} + \frac{18}{41} a^{9} - \frac{5}{41} a^{8} + \frac{8}{41} a^{7} + \frac{12}{41} a^{6} - \frac{7}{41} a^{5} - \frac{13}{41} a^{4} - \frac{9}{41} a^{3} - \frac{37}{82} a^{2} + \frac{4}{41} a$, $\frac{1}{82} a^{18} + \frac{5}{82} a^{15} - \frac{1}{82} a^{14} + \frac{1}{41} a^{13} + \frac{3}{82} a^{12} + \frac{17}{82} a^{11} - \frac{18}{41} a^{10} + \frac{1}{82} a^{9} + \frac{3}{82} a^{8} - \frac{8}{41} a^{7} - \frac{35}{82} a^{6} - \frac{5}{82} a^{5} + \frac{2}{41} a^{4} + \frac{12}{41} a^{3} + \frac{37}{82} a^{2} - \frac{2}{41} a$, $\frac{1}{4209058958905942} a^{19} + \frac{5181789051123}{2104529479452971} a^{18} - \frac{4881190440462}{2104529479452971} a^{17} + \frac{6992884106214}{2104529479452971} a^{16} + \frac{5291132060602}{2104529479452971} a^{15} + \frac{831524133544345}{4209058958905942} a^{14} - \frac{441932048162739}{2104529479452971} a^{13} + \frac{228801678421616}{2104529479452971} a^{12} + \frac{527085657019101}{4209058958905942} a^{11} + \frac{92185739687670}{2104529479452971} a^{10} + \frac{471129549582272}{2104529479452971} a^{9} - \frac{754333434720563}{4209058958905942} a^{8} + \frac{1023603737705178}{2104529479452971} a^{7} + \frac{875510078945917}{2104529479452971} a^{6} - \frac{560049381089701}{4209058958905942} a^{5} - \frac{1958461313285227}{4209058958905942} a^{4} - \frac{3909995314868}{51329987303731} a^{3} - \frac{1329789727243205}{4209058958905942} a^{2} + \frac{519021938986836}{2104529479452971} a + \frac{13513494645516}{51329987303731}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{84391488155618}{2104529479452971} a^{19} + \frac{356636710342078}{2104529479452971} a^{18} - \frac{2755075330566425}{4209058958905942} a^{17} + \frac{7590728758339177}{4209058958905942} a^{16} - \frac{8055597930620912}{2104529479452971} a^{15} + \frac{14917545240599615}{2104529479452971} a^{14} - \frac{21103533052420195}{2104529479452971} a^{13} + \frac{25714059074873476}{2104529479452971} a^{12} - \frac{24659887357467279}{2104529479452971} a^{11} + \frac{19041177257029201}{2104529479452971} a^{10} - \frac{14224130368815338}{2104529479452971} a^{9} + \frac{8338472659137343}{2104529479452971} a^{8} - \frac{8355259610613490}{2104529479452971} a^{7} - \frac{6824839866858914}{2104529479452971} a^{6} - \frac{624544470362671}{2104529479452971} a^{5} - \frac{5712297983939406}{2104529479452971} a^{4} - \frac{1475395151637769}{2104529479452971} a^{3} - \frac{1483153337487647}{4209058958905942} a^{2} + \frac{6780722042203047}{4209058958905942} a + \frac{67741716644540}{51329987303731} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 21670.1464682 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.253125.1 x5, 10.2.320361328125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.253125.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
3Data not computed
5Data not computed