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Properties

Label 20.4.16048707324...0625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{18}\cdot 29^{10}$
Root discriminant $22.92$
Ramified primes $5, 29$
Class number $2$
Class group $[2]$
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![-1, 4, 7, -59, 78, 151, -501, 612, -529, 433, -598, 1075, -1375, 1330, -955, 497, -178, 31, 8, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 8*x^18 + 31*x^17 - 178*x^16 + 497*x^15 - 955*x^14 + 1330*x^13 - 1375*x^12 + 1075*x^11 - 598*x^10 + 433*x^9 - 529*x^8 + 612*x^7 - 501*x^6 + 151*x^5 + 78*x^4 - 59*x^3 + 7*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 8*x^18 + 31*x^17 - 178*x^16 + 497*x^15 - 955*x^14 + 1330*x^13 - 1375*x^12 + 1075*x^11 - 598*x^10 + 433*x^9 - 529*x^8 + 612*x^7 - 501*x^6 + 151*x^5 + 78*x^4 - 59*x^3 + 7*x^2 + 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 8 x^{18} + 31 x^{17} - 178 x^{16} + 497 x^{15} - 955 x^{14} + 1330 x^{13} - 1375 x^{12} + 1075 x^{11} - 598 x^{10} + 433 x^{9} - 529 x^{8} + 612 x^{7} - 501 x^{6} + 151 x^{5} + 78 x^{4} - 59 x^{3} + 7 x^{2} + 4 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1604870732498935699462890625=5^{18}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.92$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{12} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{10} + \frac{1}{5} a^{5} - \frac{1}{5}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{11} + \frac{1}{5} a^{6} - \frac{1}{5} a$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{12} + \frac{1}{5} a^{7} - \frac{1}{5} a^{2}$, $\frac{1}{95} a^{18} - \frac{2}{95} a^{17} + \frac{9}{95} a^{16} - \frac{8}{95} a^{15} + \frac{4}{95} a^{14} + \frac{4}{95} a^{13} + \frac{28}{95} a^{12} - \frac{4}{95} a^{11} - \frac{18}{95} a^{10} + \frac{36}{95} a^{9} - \frac{8}{95} a^{8} + \frac{22}{95} a^{7} + \frac{2}{5} a^{6} - \frac{7}{95} a^{5} - \frac{7}{19} a^{4} + \frac{43}{95} a^{3} - \frac{8}{95} a^{2} - \frac{8}{95} a - \frac{2}{95}$, $\frac{1}{347895372587725} a^{19} - \frac{13023460089}{69579074517545} a^{18} - \frac{31672746268002}{347895372587725} a^{17} - \frac{28698870507796}{347895372587725} a^{16} + \frac{74774386761}{347895372587725} a^{15} - \frac{17648482087457}{347895372587725} a^{14} - \frac{11722307391802}{347895372587725} a^{13} - \frac{6582667427147}{347895372587725} a^{12} - \frac{152572354232117}{347895372587725} a^{11} - \frac{1770729037987}{347895372587725} a^{10} + \frac{111896678504}{13915814903509} a^{9} - \frac{1450288639268}{18310282767775} a^{8} + \frac{15766897049384}{347895372587725} a^{7} + \frac{158995511127356}{347895372587725} a^{6} - \frac{24590641609701}{69579074517545} a^{5} + \frac{60415215581271}{347895372587725} a^{4} + \frac{18474209085754}{347895372587725} a^{3} + \frac{9469752842986}{69579074517545} a^{2} + \frac{158416794421947}{347895372587725} a + \frac{20472279044596}{347895372587725}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 406208.828043 \)
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

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 10.2.8012167578125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{10}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$29$29.10.5.1$x^{10} - 1682 x^{6} + 707281 x^{2} - 2481849029$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
29.10.5.1$x^{10} - 1682 x^{6} + 707281 x^{2} - 2481849029$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$