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Properties

Label 20.0.50955348553...0689.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{19}\cdot 547^{9}$
Root discriminant $48.46$
Ramified primes $3, 547$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:S_4$ (as 20T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![87, -321, 1348, -20458, 84143, -108261, 53197, 11779, -7296, -11239, 4106, 5534, -1828, -1060, 306, -29, 124, -60, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 60*x^17 + 124*x^16 - 29*x^15 + 306*x^14 - 1060*x^13 - 1828*x^12 + 5534*x^11 + 4106*x^10 - 11239*x^9 - 7296*x^8 + 11779*x^7 + 53197*x^6 - 108261*x^5 + 84143*x^4 - 20458*x^3 + 1348*x^2 - 321*x + 87)
 
gp: K = bnfinit(x^20 - x^19 + 5*x^18 - 60*x^17 + 124*x^16 - 29*x^15 + 306*x^14 - 1060*x^13 - 1828*x^12 + 5534*x^11 + 4106*x^10 - 11239*x^9 - 7296*x^8 + 11779*x^7 + 53197*x^6 - 108261*x^5 + 84143*x^4 - 20458*x^3 + 1348*x^2 - 321*x + 87, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 5 x^{18} - 60 x^{17} + 124 x^{16} - 29 x^{15} + 306 x^{14} - 1060 x^{13} - 1828 x^{12} + 5534 x^{11} + 4106 x^{10} - 11239 x^{9} - 7296 x^{8} + 11779 x^{7} + 53197 x^{6} - 108261 x^{5} + 84143 x^{4} - 20458 x^{3} + 1348 x^{2} - 321 x + 87 \)

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:  \(5095534855386564657677272489420689=3^{19}\cdot 547^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.46$
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:  $3, 547$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{4}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{4}{9} a^{5} - \frac{1}{3} a^{4} - \frac{4}{9} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{2}{9} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{11} - \frac{2}{9} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3}$, $\frac{1}{297} a^{18} + \frac{1}{297} a^{16} + \frac{7}{297} a^{15} - \frac{5}{297} a^{14} + \frac{4}{297} a^{13} - \frac{1}{9} a^{12} - \frac{20}{297} a^{11} - \frac{1}{9} a^{10} + \frac{4}{297} a^{9} - \frac{5}{99} a^{8} - \frac{83}{297} a^{7} - \frac{125}{297} a^{6} - \frac{125}{297} a^{5} - \frac{119}{297} a^{4} - \frac{5}{11} a^{3} + \frac{37}{297} a^{2} + \frac{13}{33} a - \frac{34}{99}$, $\frac{1}{12774816714805520556689007087960955652553} a^{19} - \frac{2003890033641369391018778714782636114}{12774816714805520556689007087960955652553} a^{18} - \frac{157891907822495297800225376761447753973}{12774816714805520556689007087960955652553} a^{17} - \frac{16723788671758635254581485207933040508}{473141359807611872469963225480035394539} a^{16} - \frac{82688322671723326257817823322090379051}{4258272238268506852229669029320318550851} a^{15} - \frac{32716560790109686460923137974774225354}{4258272238268506852229669029320318550851} a^{14} - \frac{662427018717628561676598861439912288963}{12774816714805520556689007087960955652553} a^{13} - \frac{1334322113333995393719865967706949501142}{12774816714805520556689007087960955652553} a^{12} + \frac{1412149679303859118284543151601626901960}{12774816714805520556689007087960955652553} a^{11} + \frac{809478159495188996206110232401868932184}{12774816714805520556689007087960955652553} a^{10} + \frac{109804394093926303700966757028267556227}{1161346974073229141517182462541905059323} a^{9} + \frac{5747925101136800851273622847342242344678}{12774816714805520556689007087960955652553} a^{8} - \frac{785953981888761653782468257781263051181}{4258272238268506852229669029320318550851} a^{7} + \frac{100869939000220764093378140102134022909}{1419424079422835617409889676440106183617} a^{6} - \frac{514851061429075343413121703490564427495}{1419424079422835617409889676440106183617} a^{5} + \frac{1489950157104741969183468544118617075481}{12774816714805520556689007087960955652553} a^{4} + \frac{4517520798169071917098014493717234864489}{12774816714805520556689007087960955652553} a^{3} - \frac{328637682333220147998680560157892196379}{1161346974073229141517182462541905059323} a^{2} - \frac{735271568807767266428061886847483055910}{4258272238268506852229669029320318550851} a + \frac{257014010601088068734553527481200505889}{4258272238268506852229669029320318550851}$

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:  \( 3446960700.47 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:S_4$ (as 20T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 13 conjugacy class representatives for $C_5:S_4$
Character table for $C_5:S_4$

Intermediate fields

4.0.14769.1, 5.5.2692881.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
547Data not computed