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Properties

Label 20.20.4669942026...3125.1
Degree $20$
Signature $[20, 0]$
Discriminant $3^{10}\cdot 5^{31}\cdot 19^{8}$
Root discriminant $68.15$
Ramified primes $3, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:F_5$ (as 20T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6759, 65070, -172410, -110925, 1069270, -957916, -1487715, 2772445, -301350, -2101065, 1404121, 134935, -442755, 128000, 20785, -15246, 1095, 515, -75, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 75*x^18 + 515*x^17 + 1095*x^16 - 15246*x^15 + 20785*x^14 + 128000*x^13 - 442755*x^12 + 134935*x^11 + 1404121*x^10 - 2101065*x^9 - 301350*x^8 + 2772445*x^7 - 1487715*x^6 - 957916*x^5 + 1069270*x^4 - 110925*x^3 - 172410*x^2 + 65070*x - 6759)
 
gp: K = bnfinit(x^20 - 5*x^19 - 75*x^18 + 515*x^17 + 1095*x^16 - 15246*x^15 + 20785*x^14 + 128000*x^13 - 442755*x^12 + 134935*x^11 + 1404121*x^10 - 2101065*x^9 - 301350*x^8 + 2772445*x^7 - 1487715*x^6 - 957916*x^5 + 1069270*x^4 - 110925*x^3 - 172410*x^2 + 65070*x - 6759, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 75 x^{18} + 515 x^{17} + 1095 x^{16} - 15246 x^{15} + 20785 x^{14} + 128000 x^{13} - 442755 x^{12} + 134935 x^{11} + 1404121 x^{10} - 2101065 x^{9} - 301350 x^{8} + 2772445 x^{7} - 1487715 x^{6} - 957916 x^{5} + 1069270 x^{4} - 110925 x^{3} - 172410 x^{2} + 65070 x - 6759 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4669942026995164342224597930908203125=3^{10}\cdot 5^{31}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.15$
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:  $3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{15} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{7}{15} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{2}{5}$, $\frac{1}{30} a^{11} - \frac{1}{30} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{7}{30} a^{6} + \frac{13}{30} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{3}{10} a + \frac{3}{10}$, $\frac{1}{30} a^{12} - \frac{1}{30} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{13}{30} a^{7} - \frac{1}{3} a^{6} + \frac{13}{30} a^{5} + \frac{1}{3} a^{4} + \frac{11}{30} a^{2} + \frac{3}{10}$, $\frac{1}{30} a^{13} - \frac{1}{30} a^{10} - \frac{1}{10} a^{8} - \frac{1}{3} a^{6} + \frac{13}{30} a^{5} + \frac{11}{30} a^{3} - \frac{1}{3} a^{2} + \frac{3}{10}$, $\frac{1}{30} a^{14} - \frac{1}{30} a^{10} + \frac{7}{30} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{13}{30} a^{5} - \frac{3}{10} a^{4} + \frac{3}{10}$, $\frac{1}{30} a^{15} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{4}{15} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{10}$, $\frac{1}{30} a^{16} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{4}{15} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{10} a$, $\frac{1}{30} a^{17} + \frac{1}{3} a^{8} + \frac{1}{15} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{7}{30} a^{2}$, $\frac{1}{69300} a^{18} + \frac{493}{69300} a^{17} + \frac{379}{34650} a^{16} - \frac{17}{3150} a^{15} - \frac{193}{13860} a^{14} + \frac{439}{34650} a^{13} - \frac{191}{11550} a^{12} + \frac{149}{11550} a^{11} - \frac{569}{23100} a^{10} - \frac{6151}{13860} a^{9} - \frac{3151}{34650} a^{8} + \frac{757}{34650} a^{7} - \frac{3694}{17325} a^{6} - \frac{6847}{69300} a^{5} - \frac{173}{1260} a^{4} + \frac{601}{3300} a^{3} - \frac{1933}{7700} a^{2} - \frac{292}{1925} a + \frac{529}{7700}$, $\frac{1}{51736849281468901350133959918600} a^{19} - \frac{34389803938588224481893767}{12934212320367225337533489979650} a^{18} + \frac{15496119064666351404963148373}{2069473971258756054005358396744} a^{17} - \frac{18968004229862850530833752989}{1847744617195317905361927139950} a^{16} + \frac{12417590401447105787471134897}{7390978468781271621447708559800} a^{15} - \frac{270728624151181540391349563827}{51736849281468901350133959918600} a^{14} + \frac{1228061962073951120254857601}{783891655779831838638393332100} a^{13} + \frac{27300482999980651500662357011}{1724561642715630045004465330620} a^{12} + \frac{34221919612322216281694417271}{5748538809052100150014884435400} a^{11} - \frac{690220478970841724121915307129}{25868424640734450675066979959300} a^{10} + \frac{779947115947731024286010225633}{4703349934678991031830359992600} a^{9} + \frac{2825098311508692934087862187163}{25868424640734450675066979959300} a^{8} + \frac{615143416781423841842376209999}{1293421232036722533753348997965} a^{7} - \frac{17235783086352865225538641912411}{51736849281468901350133959918600} a^{6} + \frac{1043592958571669742441191699843}{3695489234390635810723854279900} a^{5} + \frac{1162823854473258139046876327311}{8622808213578150225022326653100} a^{4} - \frac{2123100131452727777349727449259}{4311404106789075112511163326550} a^{3} - \frac{482021983424925468857469963323}{3449123285431260090008930661240} a^{2} + \frac{2741469429187107411242702261837}{5748538809052100150014884435400} a + \frac{2794904583631698534113803095871}{5748538809052100150014884435400}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $19$
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:  \( 1820507440260 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:F_5$ (as 20T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 10 conjugacy class representatives for $C_5:F_5$
Character table for $C_5:F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 10.10.322143585205078125.1 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
Degree 20 sibling: data not computed
Degree 25 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{10}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.10.8.1$x^{10} - 209 x^{5} + 11552$$5$$2$$8$$D_5$$[\ ]_{5}^{2}$