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Properties

Label 20.20.3607394325...2432.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{25}\cdot 401^{10}$
Root discriminant $47.63$
Ramified primes $2, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:D_4$ (as 20T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2592, 0, -32976, 0, 125920, 0, -221656, 0, 211202, 0, -116321, 0, 37781, 0, -7195, 0, 779, 0, -44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 44*x^18 + 779*x^16 - 7195*x^14 + 37781*x^12 - 116321*x^10 + 211202*x^8 - 221656*x^6 + 125920*x^4 - 32976*x^2 + 2592)
 
gp: K = bnfinit(x^20 - 44*x^18 + 779*x^16 - 7195*x^14 + 37781*x^12 - 116321*x^10 + 211202*x^8 - 221656*x^6 + 125920*x^4 - 32976*x^2 + 2592, 1)
 

Normalized defining polynomial

\( x^{20} - 44 x^{18} + 779 x^{16} - 7195 x^{14} + 37781 x^{12} - 116321 x^{10} + 211202 x^{8} - 221656 x^{6} + 125920 x^{4} - 32976 x^{2} + 2592 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3607394325588992472373555921682432=2^{25}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.63$
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:  $2, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{24} a^{12} - \frac{1}{6} a^{10} + \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{11}{24} a^{4} - \frac{1}{24} a^{2} - \frac{1}{4}$, $\frac{1}{24} a^{13} - \frac{1}{6} a^{11} + \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{11}{24} a^{5} - \frac{1}{24} a^{3} - \frac{1}{4} a$, $\frac{1}{576} a^{14} - \frac{5}{288} a^{12} + \frac{11}{576} a^{10} - \frac{7}{192} a^{8} + \frac{247}{576} a^{6} + \frac{209}{576} a^{4} - \frac{29}{72} a^{2} - \frac{3}{16}$, $\frac{1}{576} a^{15} - \frac{5}{288} a^{13} + \frac{11}{576} a^{11} - \frac{7}{192} a^{9} + \frac{247}{576} a^{7} + \frac{209}{576} a^{5} - \frac{29}{72} a^{3} - \frac{3}{16} a$, $\frac{1}{4608} a^{16} - \frac{89}{4608} a^{12} + \frac{665}{4608} a^{10} - \frac{1115}{4608} a^{8} - \frac{67}{1536} a^{6} + \frac{641}{2304} a^{4} + \frac{257}{1152} a^{2} - \frac{15}{64}$, $\frac{1}{4608} a^{17} - \frac{89}{4608} a^{13} + \frac{665}{4608} a^{11} + \frac{421}{4608} a^{9} - \frac{67}{1536} a^{7} + \frac{641}{2304} a^{5} + \frac{257}{1152} a^{3} + \frac{19}{192} a$, $\frac{1}{190844928} a^{18} + \frac{317}{10602496} a^{16} + \frac{8807}{190844928} a^{14} - \frac{2379425}{190844928} a^{12} + \frac{3270325}{63614976} a^{10} - \frac{5712797}{63614976} a^{8} + \frac{2612293}{23855616} a^{6} - \frac{9913429}{23855616} a^{4} - \frac{2378161}{11927808} a^{2} - \frac{11899}{1325312}$, $\frac{1}{572534784} a^{19} + \frac{23561}{286267392} a^{17} + \frac{8807}{572534784} a^{15} - \frac{6065449}{572534784} a^{13} - \frac{89877337}{572534784} a^{11} - \frac{63317231}{572534784} a^{9} - \frac{5570975}{17891712} a^{7} + \frac{20579101}{71566848} a^{5} - \frac{7669055}{35783424} a^{3} + \frac{1002793}{3975936} a$

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:  $19$
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:  \( 16979295917.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T11):

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 $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{401}) \), 4.4.5145632.1, 5.5.160801.1 x5, 10.10.10368641602001.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$2$2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
401Data not computed