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Properties

Label 20.20.4859479613...0625.1
Degree $20$
Signature $[20, 0]$
Discriminant $5^{10}\cdot 17^{8}\cdot 61^{10}$
Root discriminant $54.24$
Ramified primes $5, 17, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times D_5$ (as 20T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -197, 0, 3064, 0, -18280, 0, 50170, 0, -63688, 0, 35792, 0, -9069, 0, 1034, 0, -53, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 53*x^18 + 1034*x^16 - 9069*x^14 + 35792*x^12 - 63688*x^10 + 50170*x^8 - 18280*x^6 + 3064*x^4 - 197*x^2 + 1)
 
gp: K = bnfinit(x^20 - 53*x^18 + 1034*x^16 - 9069*x^14 + 35792*x^12 - 63688*x^10 + 50170*x^8 - 18280*x^6 + 3064*x^4 - 197*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 53 x^{18} + 1034 x^{16} - 9069 x^{14} + 35792 x^{12} - 63688 x^{10} + 50170 x^{8} - 18280 x^{6} + 3064 x^{4} - 197 x^{2} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(48594796132978115111671719150390625=5^{10}\cdot 17^{8}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.24$
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:  $5, 17, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{1190} a^{14} + \frac{47}{1190} a^{12} + \frac{191}{1190} a^{10} - \frac{1}{2} a^{9} + \frac{59}{1190} a^{8} - \frac{1}{2} a^{7} - \frac{4}{595} a^{6} - \frac{1}{2} a^{5} + \frac{92}{595} a^{4} - \frac{1}{2} a^{3} + \frac{15}{238} a^{2} + \frac{11}{1190}$, $\frac{1}{1190} a^{15} + \frac{47}{1190} a^{13} + \frac{191}{1190} a^{11} - \frac{268}{595} a^{9} - \frac{1}{2} a^{8} + \frac{587}{1190} a^{7} - \frac{1}{2} a^{6} + \frac{92}{595} a^{5} - \frac{1}{2} a^{4} + \frac{15}{238} a^{3} - \frac{292}{595} a - \frac{1}{2}$, $\frac{1}{1190} a^{16} - \frac{233}{1190} a^{12} + \frac{1}{170} a^{10} - \frac{401}{1190} a^{8} - \frac{1}{2} a^{7} + \frac{8}{17} a^{6} - \frac{243}{1190} a^{4} - \frac{77}{170} a^{2} - \frac{1}{2} a + \frac{39}{595}$, $\frac{1}{1190} a^{17} - \frac{233}{1190} a^{13} + \frac{1}{170} a^{11} - \frac{401}{1190} a^{9} - \frac{1}{2} a^{8} + \frac{8}{17} a^{7} - \frac{243}{1190} a^{5} - \frac{77}{170} a^{3} - \frac{1}{2} a^{2} + \frac{39}{595} a$, $\frac{1}{22956290} a^{18} + \frac{8207}{22956290} a^{16} + \frac{228}{2295629} a^{14} + \frac{1801566}{11478145} a^{12} - \frac{1918407}{11478145} a^{10} - \frac{1}{2} a^{9} + \frac{681873}{4591258} a^{8} - \frac{5711977}{22956290} a^{6} - \frac{1}{2} a^{5} + \frac{273918}{675185} a^{4} - \frac{1}{2} a^{3} - \frac{930918}{2295629} a^{2} - \frac{3836361}{22956290}$, $\frac{1}{22956290} a^{19} + \frac{8207}{22956290} a^{17} + \frac{228}{2295629} a^{15} + \frac{1801566}{11478145} a^{13} - \frac{1918407}{11478145} a^{11} - \frac{806878}{2295629} a^{9} + \frac{2883084}{11478145} a^{7} - \frac{1}{2} a^{6} + \frac{273918}{675185} a^{5} - \frac{1}{2} a^{4} - \frac{930918}{2295629} a^{3} + \frac{3820892}{11478145} a - \frac{1}{2}$

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:  \( 123847275257 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_5$ (as 20T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 16 conjugacy class representatives for $C_2^2\times D_5$
Character table for $C_2^2\times D_5$

Intermediate fields

\(\Q(\sqrt{305}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{5}, \sqrt{61})\), 5.5.26884225.1, 10.10.220442273924440625.1, 10.10.3613807769253125.1, 10.10.44088454784888125.1

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 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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61Data not computed