↑

Properties

Label 20.4.11349417630...5625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 7^{8}\cdot 17^{10}$
Root discriminant $20.08$
Ramified primes $5, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times D_5$ (as 20T8)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -1, 12, -8, -27, -35, -36, -106, -33, -23, -226, 158, -169, 127, -65, 27, -14, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 - 14*x^17 + 27*x^16 - 65*x^15 + 127*x^14 - 169*x^13 + 158*x^12 - 226*x^11 - 23*x^10 - 33*x^9 - 106*x^8 - 36*x^7 - 35*x^6 - 27*x^5 - 8*x^4 + 12*x^3 - x^2 + 4*x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 5*x^18 - 14*x^17 + 27*x^16 - 65*x^15 + 127*x^14 - 169*x^13 + 158*x^12 - 226*x^11 - 23*x^10 - 33*x^9 - 106*x^8 - 36*x^7 - 35*x^6 - 27*x^5 - 8*x^4 + 12*x^3 - x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 5 x^{18} - 14 x^{17} + 27 x^{16} - 65 x^{15} + 127 x^{14} - 169 x^{13} + 158 x^{12} - 226 x^{11} - 23 x^{10} - 33 x^{9} - 106 x^{8} - 36 x^{7} - 35 x^{6} - 27 x^{5} - 8 x^{4} + 12 x^{3} - x^{2} + 4 x + 1 \)

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:  \(113494176301780230947265625=5^{10}\cdot 7^{8}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.08$
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:  $5, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{247} a^{16} + \frac{67}{247} a^{14} + \frac{10}{247} a^{13} - \frac{4}{19} a^{12} + \frac{92}{247} a^{11} + \frac{3}{19} a^{10} - \frac{40}{247} a^{9} - \frac{120}{247} a^{8} + \frac{94}{247} a^{7} + \frac{45}{247} a^{6} - \frac{84}{247} a^{5} - \frac{6}{13} a^{4} - \frac{67}{247} a^{3} - \frac{8}{19} a^{2} + \frac{6}{13} a + \frac{73}{247}$, $\frac{1}{247} a^{17} + \frac{67}{247} a^{15} + \frac{10}{247} a^{14} - \frac{4}{19} a^{13} + \frac{92}{247} a^{12} + \frac{3}{19} a^{11} - \frac{40}{247} a^{10} - \frac{120}{247} a^{9} + \frac{94}{247} a^{8} + \frac{45}{247} a^{7} - \frac{84}{247} a^{6} - \frac{6}{13} a^{5} - \frac{67}{247} a^{4} - \frac{8}{19} a^{3} + \frac{6}{13} a^{2} + \frac{73}{247} a$, $\frac{1}{4693} a^{18} + \frac{9}{4693} a^{17} + \frac{5}{4693} a^{16} - \frac{1857}{4693} a^{15} + \frac{2306}{4693} a^{14} + \frac{239}{4693} a^{13} + \frac{386}{4693} a^{12} - \frac{1935}{4693} a^{11} + \frac{1301}{4693} a^{10} + \frac{1741}{4693} a^{9} - \frac{67}{4693} a^{8} - \frac{1802}{4693} a^{7} + \frac{539}{4693} a^{6} - \frac{331}{4693} a^{5} - \frac{2037}{4693} a^{4} + \frac{368}{4693} a^{3} - \frac{357}{4693} a^{2} + \frac{505}{4693} a - \frac{1809}{4693}$, $\frac{1}{35140023829391} a^{19} + \frac{185261880}{1849474938389} a^{18} - \frac{1511628151}{1849474938389} a^{17} + \frac{17511810463}{35140023829391} a^{16} - \frac{26314215296}{142267302953} a^{15} + \frac{577408346901}{2703078756107} a^{14} - \frac{8037665121123}{35140023829391} a^{13} - \frac{1034548587078}{35140023829391} a^{12} + \frac{4722976257897}{35140023829391} a^{11} + \frac{10246092147010}{35140023829391} a^{10} - \frac{2111476404239}{35140023829391} a^{9} - \frac{165774372471}{423373781077} a^{8} - \frac{16966688773845}{35140023829391} a^{7} + \frac{1682105958113}{35140023829391} a^{6} - \frac{8428965278755}{35140023829391} a^{5} + \frac{16543508006229}{35140023829391} a^{4} + \frac{14436616117516}{35140023829391} a^{3} + \frac{4236815101646}{35140023829391} a^{2} - \frac{6497163262333}{35140023829391} a - \frac{11801388893410}{35140023829391}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 106567.833621 \) (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

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 5.1.14161.1, 10.2.626668503125.1, 10.2.10653364553125.1, 10.2.3409076657.1

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 R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.4.2.1$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}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$