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Properties

Label 20.0.77299548986...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{25}\cdot 11^{10}$
Root discriminant $24.80$
Ramified primes $5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{20}$ (as 20T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -255, 550, -1055, 1445, -1714, 1810, -1330, 810, -315, -39, 45, 160, -130, 120, -79, 25, -5, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^17 + 25*x^16 - 79*x^15 + 120*x^14 - 130*x^13 + 160*x^12 + 45*x^11 - 39*x^10 - 315*x^9 + 810*x^8 - 1330*x^7 + 1810*x^6 - 1714*x^5 + 1445*x^4 - 1055*x^3 + 550*x^2 - 255*x + 81)
 
gp: K = bnfinit(x^20 - 5*x^17 + 25*x^16 - 79*x^15 + 120*x^14 - 130*x^13 + 160*x^12 + 45*x^11 - 39*x^10 - 315*x^9 + 810*x^8 - 1330*x^7 + 1810*x^6 - 1714*x^5 + 1445*x^4 - 1055*x^3 + 550*x^2 - 255*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{17} + 25 x^{16} - 79 x^{15} + 120 x^{14} - 130 x^{13} + 160 x^{12} + 45 x^{11} - 39 x^{10} - 315 x^{9} + 810 x^{8} - 1330 x^{7} + 1810 x^{6} - 1714 x^{5} + 1445 x^{4} - 1055 x^{3} + 550 x^{2} - 255 x + 81 \)

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:  \(7729954898655414581298828125=5^{25}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.80$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{15} a^{10} + \frac{1}{5} a^{5} + \frac{1}{3} a^{2} + \frac{2}{5}$, $\frac{1}{15} a^{11} + \frac{1}{5} a^{6} + \frac{1}{3} a^{3} + \frac{2}{5} a$, $\frac{1}{15} a^{12} + \frac{1}{5} a^{7} + \frac{1}{3} a^{4} + \frac{2}{5} a^{2}$, $\frac{1}{45} a^{13} - \frac{1}{45} a^{10} - \frac{1}{9} a^{9} - \frac{2}{45} a^{8} - \frac{1}{9} a^{7} - \frac{4}{9} a^{6} - \frac{2}{5} a^{5} + \frac{2}{9} a^{4} - \frac{14}{45} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a + \frac{1}{5}$, $\frac{1}{45} a^{14} - \frac{1}{45} a^{11} + \frac{1}{45} a^{10} - \frac{2}{45} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{2}{5} a^{6} - \frac{17}{45} a^{5} - \frac{14}{45} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{225} a^{15} + \frac{1}{45} a^{12} - \frac{1}{45} a^{11} + \frac{7}{225} a^{10} - \frac{1}{45} a^{9} - \frac{1}{45} a^{8} + \frac{4}{15} a^{7} - \frac{4}{45} a^{6} - \frac{62}{225} a^{5} - \frac{14}{45} a^{4} - \frac{1}{5} a^{3} + \frac{4}{15} a^{2} - \frac{1}{3} a + \frac{6}{25}$, $\frac{1}{2475} a^{16} + \frac{1}{495} a^{15} + \frac{1}{99} a^{14} - \frac{4}{495} a^{13} - \frac{8}{495} a^{12} + \frac{17}{2475} a^{11} - \frac{14}{495} a^{10} + \frac{1}{55} a^{9} - \frac{53}{495} a^{8} - \frac{2}{99} a^{7} - \frac{157}{2475} a^{6} + \frac{109}{495} a^{5} + \frac{101}{495} a^{4} - \frac{163}{495} a^{3} + \frac{203}{495} a^{2} + \frac{21}{275} a - \frac{9}{55}$, $\frac{1}{2475} a^{17} + \frac{4}{495} a^{14} + \frac{1}{495} a^{13} + \frac{52}{2475} a^{12} + \frac{2}{495} a^{11} - \frac{1}{55} a^{10} + \frac{56}{495} a^{9} - \frac{53}{495} a^{8} + \frac{698}{2475} a^{7} - \frac{5}{33} a^{6} + \frac{13}{55} a^{5} - \frac{17}{99} a^{4} + \frac{17}{495} a^{3} - \frac{1201}{2475} a^{2} - \frac{79}{165} a + \frac{23}{55}$, $\frac{1}{423225} a^{18} - \frac{17}{141075} a^{17} + \frac{23}{423225} a^{16} - \frac{173}{423225} a^{15} - \frac{28}{7695} a^{14} + \frac{1537}{423225} a^{13} - \frac{6092}{423225} a^{12} + \frac{12706}{423225} a^{11} - \frac{1222}{141075} a^{10} - \frac{524}{84645} a^{9} - \frac{524}{12825} a^{8} + \frac{127327}{423225} a^{7} + \frac{62108}{141075} a^{6} + \frac{157846}{423225} a^{5} + \frac{16817}{84645} a^{4} - \frac{99521}{423225} a^{3} - \frac{157709}{423225} a^{2} - \frac{54676}{141075} a + \frac{953}{5225}$, $\frac{1}{25816725} a^{19} - \frac{1}{8605575} a^{18} + \frac{2876}{25816725} a^{17} + \frac{17}{271755} a^{16} - \frac{2978}{2346975} a^{15} + \frac{50737}{25816725} a^{14} + \frac{20999}{2346975} a^{13} + \frac{62632}{25816725} a^{12} + \frac{12191}{573705} a^{11} + \frac{328769}{25816725} a^{10} - \frac{11434}{782325} a^{9} + \frac{504571}{25816725} a^{8} - \frac{252443}{2868525} a^{7} + \frac{1344797}{5163345} a^{6} - \frac{10679399}{25816725} a^{5} + \frac{7352284}{25816725} a^{4} + \frac{7579918}{25816725} a^{3} + \frac{458252}{956175} a^{2} + \frac{32374}{573705} a + \frac{46523}{318725}$

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:  $9$
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:  \( 2264798.97636 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{20}$ (as 20T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $D_{20}$
Character table for $D_{20}$

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.605.1, 5.1.1890625.1, 10.0.39319091796875.1

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

Sibling fields

Galois closure: data not computed
Degree 20 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 $20$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R $20$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.10.13.8$x^{10} + 20 x^{4} + 10$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$
5.10.12.7$x^{10} + 10 x^{8} + 5 x^{7} + 15 x^{6} + 5 x^{4} + 5 x^{2} - 20 x + 7$$5$$2$$12$$D_{10}$$[3/2]_{2}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$