Properties

Label 20.0.31059261593...3401.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 47^{10}$
Root discriminant $11.87$
Ramified primes $3, 47$
Class number $1$
Class group Trivial
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -5, 0, 22, 1, -38, -11, 46, 11, -51, 11, 46, -11, -38, 1, 22, 0, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 5 x^{18} + 22 x^{16} + x^{15} - 38 x^{14} - 11 x^{13} + 46 x^{12} + 11 x^{11} - 51 x^{10} + 11 x^{9} + 46 x^{8} - 11 x^{7} - 38 x^{6} + x^{5} + 22 x^{4} - 5 x^{2} - x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3105926159393528563401=3^{10}\cdot 47^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.87$
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:  $3, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Gal(K/\Q)|$:  $20$
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{15} - \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{158455} a^{18} - \frac{7114}{158455} a^{17} - \frac{8651}{158455} a^{16} + \frac{61137}{158455} a^{15} + \frac{68476}{158455} a^{14} + \frac{23128}{158455} a^{13} - \frac{38306}{158455} a^{12} + \frac{5754}{14405} a^{11} - \frac{243}{31691} a^{10} - \frac{72631}{158455} a^{9} - \frac{243}{31691} a^{8} + \frac{5754}{14405} a^{7} - \frac{38306}{158455} a^{6} + \frac{23128}{158455} a^{5} + \frac{68476}{158455} a^{4} + \frac{61137}{158455} a^{3} - \frac{8651}{158455} a^{2} - \frac{7114}{158455} a + \frac{1}{158455}$, $\frac{1}{158455} a^{19} - \frac{1424}{31691} a^{17} - \frac{1537}{158455} a^{16} + \frac{13962}{31691} a^{15} + \frac{1468}{31691} a^{14} - \frac{4126}{14405} a^{13} - \frac{12289}{31691} a^{12} - \frac{56809}{158455} a^{11} - \frac{64498}{158455} a^{10} - \frac{6497}{14405} a^{9} + \frac{71427}{158455} a^{8} + \frac{12911}{31691} a^{7} + \frac{56844}{158455} a^{6} + \frac{61396}{158455} a^{5} + \frac{45349}{158455} a^{4} - \frac{7317}{158455} a^{3} - \frac{69788}{158455} a^{2} + \frac{1532}{158455} a + \frac{7114}{158455}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{32919}{31691} a^{19} + \frac{45216}{158455} a^{18} + \frac{867594}{158455} a^{17} + \frac{625479}{158455} a^{16} - \frac{3248754}{158455} a^{15} - \frac{230742}{14405} a^{14} + \frac{943475}{31691} a^{13} + \frac{5405924}{158455} a^{12} - \frac{4343186}{158455} a^{11} - \frac{5238968}{158455} a^{10} + \frac{5266152}{158455} a^{9} + \frac{37736}{2365} a^{8} - \frac{99718}{2365} a^{7} - \frac{3087386}{158455} a^{6} + \frac{984228}{31691} a^{5} + \frac{3353628}{158455} a^{4} - \frac{1884874}{158455} a^{3} - \frac{149566}{14405} a^{2} + \frac{192919}{158455} a + \frac{415746}{158455} \) (order $6$)
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:  \( 630.365304677 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{141}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-47})\), 5.1.2209.1 x5, 10.0.229345007.1, 10.2.55730836701.1 x5, 10.0.1185762483.1 x5

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

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$