↑

Properties

Label 20.2.10929603344...9968.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{40}\cdot 1583^{5}$
Root discriminant $25.23$
Ramified primes $2, 1583$
Class number $1$
Class group Trivial
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-199, 600, 56, -1504, 596, 2060, -3080, 1392, 1615, -2160, 412, 1604, -2062, 1376, -474, -16, 134, -88, 34, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 34*x^18 - 88*x^17 + 134*x^16 - 16*x^15 - 474*x^14 + 1376*x^13 - 2062*x^12 + 1604*x^11 + 412*x^10 - 2160*x^9 + 1615*x^8 + 1392*x^7 - 3080*x^6 + 2060*x^5 + 596*x^4 - 1504*x^3 + 56*x^2 + 600*x - 199)
 
gp: K = bnfinit(x^20 - 8*x^19 + 34*x^18 - 88*x^17 + 134*x^16 - 16*x^15 - 474*x^14 + 1376*x^13 - 2062*x^12 + 1604*x^11 + 412*x^10 - 2160*x^9 + 1615*x^8 + 1392*x^7 - 3080*x^6 + 2060*x^5 + 596*x^4 - 1504*x^3 + 56*x^2 + 600*x - 199, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 34 x^{18} - 88 x^{17} + 134 x^{16} - 16 x^{15} - 474 x^{14} + 1376 x^{13} - 2062 x^{12} + 1604 x^{11} + 412 x^{10} - 2160 x^{9} + 1615 x^{8} + 1392 x^{7} - 3080 x^{6} + 2060 x^{5} + 596 x^{4} - 1504 x^{3} + 56 x^{2} + 600 x - 199 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-10929603344110461118702419968=-\,2^{40}\cdot 1583^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.23$
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:  $2, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{73} a^{16} - \frac{10}{73} a^{15} - \frac{11}{73} a^{14} + \frac{21}{73} a^{13} - \frac{30}{73} a^{12} - \frac{27}{73} a^{11} + \frac{35}{73} a^{10} - \frac{5}{73} a^{9} - \frac{27}{73} a^{8} + \frac{10}{73} a^{7} + \frac{26}{73} a^{6} - \frac{2}{73} a^{5} - \frac{14}{73} a^{4} - \frac{13}{73} a^{3} - \frac{19}{73} a^{2} + \frac{34}{73} a - \frac{25}{73}$, $\frac{1}{73} a^{17} + \frac{35}{73} a^{15} - \frac{16}{73} a^{14} + \frac{34}{73} a^{13} - \frac{35}{73} a^{12} - \frac{16}{73} a^{11} - \frac{20}{73} a^{10} - \frac{4}{73} a^{9} + \frac{32}{73} a^{8} - \frac{20}{73} a^{7} - \frac{34}{73} a^{6} - \frac{34}{73} a^{5} - \frac{7}{73} a^{4} - \frac{3}{73} a^{3} - \frac{10}{73} a^{2} + \frac{23}{73} a - \frac{31}{73}$, $\frac{1}{73} a^{18} - \frac{31}{73} a^{15} - \frac{19}{73} a^{14} + \frac{33}{73} a^{13} + \frac{12}{73} a^{12} - \frac{24}{73} a^{11} + \frac{12}{73} a^{10} - \frac{12}{73} a^{9} - \frac{24}{73} a^{8} - \frac{19}{73} a^{7} + \frac{5}{73} a^{6} - \frac{10}{73} a^{5} - \frac{24}{73} a^{4} + \frac{7}{73} a^{3} + \frac{31}{73} a^{2} + \frac{20}{73} a - \frac{1}{73}$, $\frac{1}{126495109514247652999123547} a^{19} + \frac{38167854397686700387317}{126495109514247652999123547} a^{18} - \frac{776127948876987536678814}{126495109514247652999123547} a^{17} + \frac{464383456011940176546145}{126495109514247652999123547} a^{16} - \frac{6538296897208385328656584}{126495109514247652999123547} a^{15} - \frac{59569206019160137358302172}{126495109514247652999123547} a^{14} - \frac{18386055318096531300919283}{126495109514247652999123547} a^{13} + \frac{39154674898650239916621866}{126495109514247652999123547} a^{12} - \frac{23441717261848473628915110}{126495109514247652999123547} a^{11} - \frac{7919553236700226910965174}{126495109514247652999123547} a^{10} + \frac{37808701142579434578912415}{126495109514247652999123547} a^{9} + \frac{43965066741464530116722004}{126495109514247652999123547} a^{8} + \frac{62572032855663621408068900}{126495109514247652999123547} a^{7} + \frac{19399810940529909484001250}{126495109514247652999123547} a^{6} - \frac{50529761478667278246734676}{126495109514247652999123547} a^{5} + \frac{8463361182268319773703920}{126495109514247652999123547} a^{4} - \frac{54160054537152733370230596}{126495109514247652999123547} a^{3} + \frac{53302810835596149921811169}{126495109514247652999123547} a^{2} - \frac{45414369790384318807904979}{126495109514247652999123547} a - \frac{31921213854377846933921344}{126495109514247652999123547}$

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:  $10$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 854898.491998 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.405248.1, 10.6.82112970752.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
2Data not computed
1583Data not computed