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Properties

Label 20.0.46566128730...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{31}$
Root discriminant $12.12$
Ramified prime $5$
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![1, 0, 0, -15, 65, -128, 175, -285, 585, -925, 879, -375, -35, -85, 445, -552, 375, -165, 50, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 50*x^18 - 165*x^17 + 375*x^16 - 552*x^15 + 445*x^14 - 85*x^13 - 35*x^12 - 375*x^11 + 879*x^10 - 925*x^9 + 585*x^8 - 285*x^7 + 175*x^6 - 128*x^5 + 65*x^4 - 15*x^3 + 1)
 
gp: K = bnfinit(x^20 - 10*x^19 + 50*x^18 - 165*x^17 + 375*x^16 - 552*x^15 + 445*x^14 - 85*x^13 - 35*x^12 - 375*x^11 + 879*x^10 - 925*x^9 + 585*x^8 - 285*x^7 + 175*x^6 - 128*x^5 + 65*x^4 - 15*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 50 x^{18} - 165 x^{17} + 375 x^{16} - 552 x^{15} + 445 x^{14} - 85 x^{13} - 35 x^{12} - 375 x^{11} + 879 x^{10} - 925 x^{9} + 585 x^{8} - 285 x^{7} + 175 x^{6} - 128 x^{5} + 65 x^{4} - 15 x^{3} + 1 \)

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:  \(4656612873077392578125=5^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.12$
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:  $5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{6} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{72} a^{16} + \frac{1}{18} a^{15} + \frac{5}{72} a^{14} + \frac{1}{8} a^{13} + \frac{1}{72} a^{12} - \frac{31}{72} a^{11} + \frac{35}{72} a^{10} - \frac{5}{24} a^{9} - \frac{3}{8} a^{8} + \frac{19}{72} a^{7} - \frac{29}{72} a^{6} - \frac{25}{72} a^{5} + \frac{1}{8} a^{4} + \frac{25}{72} a^{3} - \frac{7}{24} a^{2} + \frac{7}{18} a - \frac{13}{72}$, $\frac{1}{72} a^{17} + \frac{1}{72} a^{15} + \frac{1}{72} a^{14} + \frac{1}{72} a^{13} + \frac{1}{72} a^{12} + \frac{1}{24} a^{11} + \frac{25}{72} a^{10} - \frac{1}{24} a^{9} + \frac{19}{72} a^{8} + \frac{1}{24} a^{7} + \frac{7}{72} a^{6} + \frac{1}{72} a^{5} - \frac{23}{72} a^{4} + \frac{35}{72} a^{3} + \frac{1}{18} a^{2} + \frac{7}{72} a + \frac{1}{18}$, $\frac{1}{2736} a^{18} - \frac{1}{304} a^{17} + \frac{1}{171} a^{16} + \frac{1}{36} a^{15} + \frac{127}{2736} a^{14} - \frac{197}{2736} a^{13} + \frac{21}{304} a^{12} - \frac{671}{2736} a^{11} + \frac{13}{304} a^{10} - \frac{431}{2736} a^{9} + \frac{61}{912} a^{8} - \frac{1139}{2736} a^{7} + \frac{1027}{2736} a^{6} + \frac{565}{2736} a^{5} - \frac{691}{2736} a^{4} + \frac{121}{684} a^{3} + \frac{67}{684} a^{2} + \frac{85}{2736} a - \frac{1}{912}$, $\frac{1}{1151856} a^{19} + \frac{67}{383952} a^{18} + \frac{1669}{287964} a^{17} + \frac{172}{71991} a^{16} + \frac{88477}{1151856} a^{15} - \frac{49565}{1151856} a^{14} - \frac{1371}{127984} a^{13} + \frac{174565}{1151856} a^{12} - \frac{20267}{127984} a^{11} + \frac{322477}{1151856} a^{10} - \frac{16813}{127984} a^{9} - \frac{427487}{1151856} a^{8} - \frac{231893}{1151856} a^{7} - \frac{507551}{1151856} a^{6} + \frac{505673}{1151856} a^{5} + \frac{1145}{143982} a^{4} + \frac{251765}{575928} a^{3} - \frac{71999}{1151856} a^{2} + \frac{24745}{127984} a + \frac{19085}{191976}$

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:  \( -\frac{5033}{63992} a^{19} + \frac{379501}{383952} a^{18} - \frac{6457189}{1151856} a^{17} + \frac{11464165}{575928} a^{16} - \frac{27728233}{575928} a^{15} + \frac{28478797}{383952} a^{14} - \frac{62073337}{1151856} a^{13} - \frac{20410439}{1151856} a^{12} + \frac{50215045}{1151856} a^{11} + \frac{17602247}{383952} a^{10} - \frac{18254167}{127984} a^{9} + \frac{136233611}{1151856} a^{8} - \frac{33248743}{1151856} a^{7} - \frac{8896925}{1151856} a^{6} - \frac{1656601}{383952} a^{5} + \frac{9305843}{1151856} a^{4} + \frac{367489}{191976} a^{3} - \frac{853709}{143982} a^{2} + \frac{2037535}{1151856} a + \frac{52397}{383952} \) (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:  \( 1543.06769005 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.78125.1 x5, 10.2.30517578125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.78125.1
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed