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Properties

Label 20.0.267...376.1
Degree $20$
Signature $[0, 10]$
Discriminant $2.676\times 10^{20}$
Root discriminant $10.50$
Ramified primes $2, 761$
Class number $1$
Class group trivial
Galois group $D_5\wr C_2$ (as 20T48)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 27, -54, 88, -144, 188, -140, 44, -34, 92, -92, 25, 4, 17, -18, 6, 0, 0, -2, 1]);
 

\( x^{20} - 2 x^{19} + 6 x^{16} - 18 x^{15} + 17 x^{14} + 4 x^{13} + 25 x^{12} - 92 x^{11} + 92 x^{10} - 34 x^{9} + 44 x^{8} - 140 x^{7} + 188 x^{6} - 144 x^{5} + 88 x^{4} - 54 x^{3} + 27 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(267622853577068773376\)\(\medspace = 2^{20}\cdot 761^{5}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $10.50$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 761$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $10$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{4379139123} a^{19} - \frac{119017966}{4379139123} a^{18} + \frac{1685921074}{4379139123} a^{17} - \frac{2114361461}{4379139123} a^{16} + \frac{33058990}{1459713041} a^{15} + \frac{365282}{257596419} a^{14} + \frac{114632567}{257596419} a^{13} - \frac{973911976}{4379139123} a^{12} - \frac{1115738719}{4379139123} a^{11} + \frac{537527623}{1459713041} a^{10} - \frac{379823704}{1459713041} a^{9} + \frac{493709918}{1459713041} a^{8} + \frac{230256856}{1459713041} a^{7} - \frac{1268264551}{4379139123} a^{6} + \frac{160458095}{4379139123} a^{5} + \frac{746923456}{4379139123} a^{4} - \frac{934160479}{4379139123} a^{3} - \frac{270645920}{4379139123} a^{2} + \frac{754334224}{4379139123} a + \frac{1554696913}{4379139123}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{3778880}{270869} a^{19} + \frac{11227566}{270869} a^{18} - \frac{1330507}{270869} a^{17} - \frac{6019962}{270869} a^{16} - \frac{29575666}{270869} a^{15} + \frac{83845502}{270869} a^{14} - \frac{99234599}{270869} a^{13} - \frac{28701735}{270869} a^{12} - \frac{57842432}{270869} a^{11} + \frac{495643623}{270869} a^{10} - \frac{479848985}{270869} a^{9} + \frac{109683743}{270869} a^{8} - \frac{126499126}{270869} a^{7} + \frac{696261730}{270869} a^{6} - \frac{967858891}{270869} a^{5} + \frac{644490779}{270869} a^{4} - \frac{358982021}{270869} a^{3} + \frac{241607430}{270869} a^{2} - \frac{112651583}{270869} a + \frac{21204039}{270869} \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 99.9526828541 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 99.9526828541 \cdot 1}{4\sqrt{267622853577068773376}}\approx 0.146477712569$

Galois group

$D_5\wr C_2$ (as 20T48):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.12176.1, 10.0.593019904.1 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
Degree 20 siblings: data not computed
Degree 25 sibling: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

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}$
761Data not computed