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Properties

Label 20.0.11445562015...1824.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 127^{10}$
Root discriminant $22.54$
Ramified primes $2, 127$
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![81, 0, -396, 0, 2266, 0, -1235, 0, -199, 0, 174, 0, 146, 0, -150, 0, 56, 0, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 + 56*x^16 - 150*x^14 + 146*x^12 + 174*x^10 - 199*x^8 - 1235*x^6 + 2266*x^4 - 396*x^2 + 81)
 
gp: K = bnfinit(x^20 - 11*x^18 + 56*x^16 - 150*x^14 + 146*x^12 + 174*x^10 - 199*x^8 - 1235*x^6 + 2266*x^4 - 396*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{20} - 11 x^{18} + 56 x^{16} - 150 x^{14} + 146 x^{12} + 174 x^{10} - 199 x^{8} - 1235 x^{6} + 2266 x^{4} - 396 x^{2} + 81 \)

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:  \(1144556201520286695842381824=2^{20}\cdot 127^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.54$
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:  $2, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} + \frac{1}{9} a^{8} + \frac{2}{9} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{13} + \frac{1}{9} a^{9} + \frac{2}{9} a^{5} + \frac{1}{9} a^{3}$, $\frac{1}{945} a^{16} - \frac{2}{63} a^{14} - \frac{17}{189} a^{12} + \frac{23}{189} a^{10} + \frac{121}{945} a^{8} + \frac{43}{189} a^{6} - \frac{5}{21} a^{4} - \frac{40}{189} a^{2} - \frac{31}{105}$, $\frac{1}{2835} a^{17} - \frac{1}{21} a^{15} + \frac{67}{567} a^{13} + \frac{86}{567} a^{11} + \frac{331}{2835} a^{9} - \frac{83}{567} a^{7} + \frac{34}{189} a^{5} - \frac{61}{567} a^{3} - \frac{31}{315} a$, $\frac{1}{825668235} a^{18} - \frac{4012}{18348183} a^{16} + \frac{218074}{165133647} a^{14} + \frac{19625387}{165133647} a^{12} + \frac{124909441}{825668235} a^{10} - \frac{746282}{165133647} a^{8} - \frac{2420522}{55044549} a^{6} - \frac{22973155}{165133647} a^{4} - \frac{27868051}{91740915} a^{2} - \frac{556975}{2038687}$, $\frac{1}{825668235} a^{19} + \frac{110701}{825668235} a^{17} - \frac{7645433}{165133647} a^{15} - \frac{1767335}{18348183} a^{13} - \frac{25079674}{825668235} a^{11} + \frac{30889787}{275222745} a^{9} + \frac{23609980}{165133647} a^{7} - \frac{48311122}{165133647} a^{5} + \frac{210804526}{825668235} a^{3} - \frac{3512041}{91740915} a$

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:  \( -\frac{559736}{117952605} a^{19} + \frac{6342562}{117952605} a^{17} - \frac{6512810}{23590521} a^{15} + \frac{646922}{873723} a^{13} - \frac{80454631}{117952605} a^{11} - \frac{41464861}{39317535} a^{9} + \frac{27887230}{23590521} a^{7} + \frac{155362433}{23590521} a^{5} - \frac{1373822891}{117952605} a^{3} - \frac{681937}{13105845} a \) (order $4$)
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:  \( 881125.850637 \)
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

\(\Q(\sqrt{-127}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{127}) \), \(\Q(i, \sqrt{127})\), 5.1.16129.1 x5, 10.0.33038369407.1, 10.0.266388112384.4 x5, 10.2.33831290272768.1 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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$127$127.4.2.1$x^{4} + 635 x^{2} + 145161$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
127.4.2.1$x^{4} + 635 x^{2} + 145161$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
127.4.2.1$x^{4} + 635 x^{2} + 145161$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
127.4.2.1$x^{4} + 635 x^{2} + 145161$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
127.4.2.1$x^{4} + 635 x^{2} + 145161$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$