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Properties

Label 20.4.15831129474...5449.2
Degree $20$
Signature $[4, 8]$
Discriminant $3^{10}\cdot 401^{9}$
Root discriminant $25.70$
Ramified primes $3, 401$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_2^4:D_5$ (as 20T87)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 405, 378, -729, -1926, -1620, 78, 1806, 2470, 1872, 684, -206, -439, -245, -19, 61, 33, 0, -7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 7*x^18 + 33*x^16 + 61*x^15 - 19*x^14 - 245*x^13 - 439*x^12 - 206*x^11 + 684*x^10 + 1872*x^9 + 2470*x^8 + 1806*x^7 + 78*x^6 - 1620*x^5 - 1926*x^4 - 729*x^3 + 378*x^2 + 405*x + 81)
 
gp: K = bnfinit(x^20 - x^19 - 7*x^18 + 33*x^16 + 61*x^15 - 19*x^14 - 245*x^13 - 439*x^12 - 206*x^11 + 684*x^10 + 1872*x^9 + 2470*x^8 + 1806*x^7 + 78*x^6 - 1620*x^5 - 1926*x^4 - 729*x^3 + 378*x^2 + 405*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 7 x^{18} + 33 x^{16} + 61 x^{15} - 19 x^{14} - 245 x^{13} - 439 x^{12} - 206 x^{11} + 684 x^{10} + 1872 x^{9} + 2470 x^{8} + 1806 x^{7} + 78 x^{6} - 1620 x^{5} - 1926 x^{4} - 729 x^{3} + 378 x^{2} + 405 x + 81 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15831129474510904002158875449=3^{10}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.70$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{4}{9} a^{11} - \frac{4}{9} a^{10} + \frac{4}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{1}{9} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{2}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{1}{9} a^{15} - \frac{2}{27} a^{13} - \frac{1}{27} a^{12} + \frac{4}{27} a^{11} - \frac{13}{27} a^{10} + \frac{7}{27} a^{9} + \frac{4}{9} a^{8} + \frac{2}{9} a^{7} + \frac{4}{27} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{18377149540476616177077} a^{19} + \frac{157505420643962767580}{18377149540476616177077} a^{18} + \frac{593911750704283793705}{18377149540476616177077} a^{17} + \frac{24700555924897397546}{2041905504497401797453} a^{16} - \frac{6657170387374890917}{226878389388600199717} a^{15} + \frac{1515970330036602606142}{18377149540476616177077} a^{14} + \frac{1849913843261620974977}{18377149540476616177077} a^{13} - \frac{1763214822551805071309}{18377149540476616177077} a^{12} + \frac{427845320560517575094}{18377149540476616177077} a^{11} + \frac{6356326391320986020758}{18377149540476616177077} a^{10} + \frac{287897070359350740383}{6125716513492205392359} a^{9} - \frac{1985215414708847721304}{6125716513492205392359} a^{8} + \frac{2932240676086611139396}{18377149540476616177077} a^{7} - \frac{78946956493803116917}{226878389388600199717} a^{6} - \frac{1695864639881887732274}{6125716513492205392359} a^{5} - \frac{36406466463860773339}{680635168165800599151} a^{4} + \frac{779409128098974831040}{2041905504497401797453} a^{3} + \frac{81783893558542410974}{680635168165800599151} a^{2} + \frac{104908343980019155630}{680635168165800599151} a + \frac{80607180540657786205}{226878389388600199717}$

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:  \( 1029485.32951 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T87):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.6.2094413889681.1

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 32 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{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}$
401Data not computed