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Properties

Label 20.0.59713087499...5376.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 7^{10}\cdot 17^{10}$
Root discriminant $21.82$
Ramified primes $2, 7, 17$
Class number $4$
Class group $[2, 2]$
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![1024, 0, -2816, 0, 2560, 0, -784, 0, -500, 0, 829, 0, -125, 0, -49, 0, 40, 0, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 + 40*x^16 - 49*x^14 - 125*x^12 + 829*x^10 - 500*x^8 - 784*x^6 + 2560*x^4 - 2816*x^2 + 1024)
 
gp: K = bnfinit(x^20 - 11*x^18 + 40*x^16 - 49*x^14 - 125*x^12 + 829*x^10 - 500*x^8 - 784*x^6 + 2560*x^4 - 2816*x^2 + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 11 x^{18} + 40 x^{16} - 49 x^{14} - 125 x^{12} + 829 x^{10} - 500 x^{8} - 784 x^{6} + 2560 x^{4} - 2816 x^{2} + 1024 \)

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:  \(597130874990690290159845376=2^{20}\cdot 7^{10}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.82$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{28} a^{12} - \frac{1}{4} a^{10} - \frac{5}{28} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{2}{7}$, $\frac{1}{56} a^{13} - \frac{1}{8} a^{11} - \frac{1}{2} a^{9} + \frac{23}{56} a^{7} + \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{7} a$, $\frac{1}{112} a^{14} + \frac{1}{112} a^{12} - \frac{1}{4} a^{10} - \frac{33}{112} a^{8} + \frac{23}{112} a^{6} + \frac{7}{16} a^{4} + \frac{1}{14} a^{2} - \frac{3}{7}$, $\frac{1}{224} a^{15} + \frac{1}{224} a^{13} - \frac{1}{8} a^{11} + \frac{79}{224} a^{9} + \frac{23}{224} a^{7} - \frac{9}{32} a^{5} + \frac{1}{28} a^{3} + \frac{2}{7} a$, $\frac{1}{693056} a^{16} + \frac{1489}{693056} a^{14} - \frac{83}{173264} a^{12} - \frac{64657}{693056} a^{10} + \frac{117799}{693056} a^{8} - \frac{210671}{693056} a^{6} - \frac{5305}{86632} a^{4} + \frac{1489}{10829} a^{2} - \frac{3090}{10829}$, $\frac{1}{1386112} a^{17} + \frac{1489}{1386112} a^{15} - \frac{83}{346528} a^{13} - \frac{64657}{1386112} a^{11} + \frac{117799}{1386112} a^{9} + \frac{482385}{1386112} a^{7} - \frac{5305}{173264} a^{5} - \frac{4670}{10829} a^{3} - \frac{1545}{10829} a$, $\frac{1}{47127808} a^{18} + \frac{25}{47127808} a^{16} - \frac{513}{11781952} a^{14} + \frac{520399}{47127808} a^{12} - \frac{22350817}{47127808} a^{10} + \frac{1486665}{47127808} a^{8} + \frac{12751}{105196} a^{6} + \frac{37325}{184093} a^{4} + \frac{13791}{56644} a^{2} - \frac{78814}{184093}$, $\frac{1}{94255616} a^{19} + \frac{25}{94255616} a^{17} - \frac{513}{23563904} a^{15} + \frac{520399}{94255616} a^{13} - \frac{22350817}{94255616} a^{11} - \frac{45641143}{94255616} a^{9} + \frac{12751}{210392} a^{7} - \frac{73384}{184093} a^{5} + \frac{13791}{113288} a^{3} + \frac{105279}{368186} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{2213}{517888} a^{19} - \frac{22773}{517888} a^{17} + \frac{36209}{258944} a^{15} - \frac{57237}{517888} a^{13} - \frac{319155}{517888} a^{11} + \frac{1618343}{517888} a^{9} + \frac{11603}{258944} a^{7} - \frac{217065}{64736} a^{5} + \frac{37189}{4046} a^{3} - \frac{1679}{289} a \) (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:  \( 72823.8378174 \)
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

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{119}) \), \(\Q(i, \sqrt{119})\), 5.1.14161.1 x5, 10.0.23863536599.2, 10.0.205346735104.4 x5, 10.2.24436261477376.2 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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}$
$7$7.4.2.1$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}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$