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Properties

Label 20.4.17833572791...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{8}\cdot 5^{10}\cdot 61^{10}$
Root discriminant $23.04$
Ramified primes $2, 5, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\wr C_2$ (as 20T55)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -109, 53, 170, 296, -685, -481, 2498, -3641, 3021, -1467, 414, -165, 189, -127, 34, -10, 8, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 2*x^18 + 8*x^17 - 10*x^16 + 34*x^15 - 127*x^14 + 189*x^13 - 165*x^12 + 414*x^11 - 1467*x^10 + 3021*x^9 - 3641*x^8 + 2498*x^7 - 481*x^6 - 685*x^5 + 296*x^4 + 170*x^3 + 53*x^2 - 109*x + 19)
 
gp: K = bnfinit(x^20 - 4*x^19 + 2*x^18 + 8*x^17 - 10*x^16 + 34*x^15 - 127*x^14 + 189*x^13 - 165*x^12 + 414*x^11 - 1467*x^10 + 3021*x^9 - 3641*x^8 + 2498*x^7 - 481*x^6 - 685*x^5 + 296*x^4 + 170*x^3 + 53*x^2 - 109*x + 19, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 2 x^{18} + 8 x^{17} - 10 x^{16} + 34 x^{15} - 127 x^{14} + 189 x^{13} - 165 x^{12} + 414 x^{11} - 1467 x^{10} + 3021 x^{9} - 3641 x^{8} + 2498 x^{7} - 481 x^{6} - 685 x^{5} + 296 x^{4} + 170 x^{3} + 53 x^{2} - 109 x + 19 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1783357279157206502500000000=2^{8}\cdot 5^{10}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.04$
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:  $2, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{76} a^{18} + \frac{1}{76} a^{17} - \frac{1}{19} a^{16} - \frac{1}{19} a^{15} - \frac{5}{76} a^{14} + \frac{15}{76} a^{13} - \frac{35}{76} a^{12} + \frac{5}{19} a^{11} - \frac{3}{76} a^{10} - \frac{15}{38} a^{9} - \frac{13}{38} a^{8} - \frac{7}{19} a^{7} - \frac{37}{76} a^{6} - \frac{1}{76} a^{5} - \frac{3}{76} a^{4} - \frac{6}{19} a^{3} + \frac{35}{76} a$, $\frac{1}{750290536242404962232} a^{19} + \frac{3421503567328573261}{750290536242404962232} a^{18} - \frac{35391317078720037693}{750290536242404962232} a^{17} - \frac{2054630807933488717}{14156425212120848344} a^{16} + \frac{11807767631919620625}{750290536242404962232} a^{15} - \frac{71482424844641447953}{750290536242404962232} a^{14} - \frac{10293810587408601391}{93786317030300620279} a^{13} - \frac{179916080744596907675}{750290536242404962232} a^{12} + \frac{4469478786200833375}{187572634060601240558} a^{11} - \frac{140858403240627979495}{375145268121202481116} a^{10} + \frac{129379509075859537839}{750290536242404962232} a^{9} - \frac{57317827958557227469}{187572634060601240558} a^{8} + \frac{265149619445249059323}{750290536242404962232} a^{7} + \frac{111822512182812239005}{750290536242404962232} a^{6} - \frac{22958632612816415525}{93786317030300620279} a^{5} - \frac{126937183911486504305}{750290536242404962232} a^{4} - \frac{120174320284460099921}{750290536242404962232} a^{3} - \frac{195858515788159557827}{750290536242404962232} a^{2} + \frac{117483379445596393721}{375145268121202481116} a - \frac{1301254697857933785}{39488975591705524328}$

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:  $11$
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:  \( 993656.608192 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T55):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{305}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{61})\), 10.2.42229815050000.1

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 siblings: data not computed
Degree 25 sibling: 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
61Data not computed