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Properties

Label 20.20.1505680748...0000.2
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 5^{15}\cdot 11^{16}$
Root discriminant $64.40$
Ramified primes $2, 5, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1451, 70588, -118608, -1007904, 614261, 2823768, -1203606, -3070872, 1047359, 1596028, -480740, -437336, 122474, 66220, -17726, -5524, 1435, 236, -60, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 60*x^18 + 236*x^17 + 1435*x^16 - 5524*x^15 - 17726*x^14 + 66220*x^13 + 122474*x^12 - 437336*x^11 - 480740*x^10 + 1596028*x^9 + 1047359*x^8 - 3070872*x^7 - 1203606*x^6 + 2823768*x^5 + 614261*x^4 - 1007904*x^3 - 118608*x^2 + 70588*x + 1451)
 
gp: K = bnfinit(x^20 - 4*x^19 - 60*x^18 + 236*x^17 + 1435*x^16 - 5524*x^15 - 17726*x^14 + 66220*x^13 + 122474*x^12 - 437336*x^11 - 480740*x^10 + 1596028*x^9 + 1047359*x^8 - 3070872*x^7 - 1203606*x^6 + 2823768*x^5 + 614261*x^4 - 1007904*x^3 - 118608*x^2 + 70588*x + 1451, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 60 x^{18} + 236 x^{17} + 1435 x^{16} - 5524 x^{15} - 17726 x^{14} + 66220 x^{13} + 122474 x^{12} - 437336 x^{11} - 480740 x^{10} + 1596028 x^{9} + 1047359 x^{8} - 3070872 x^{7} - 1203606 x^{6} + 2823768 x^{5} + 614261 x^{4} - 1007904 x^{3} - 118608 x^{2} + 70588 x + 1451 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1505680748169532571648000000000000000=2^{30}\cdot 5^{15}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.40$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(67,·)$, $\chi_{440}(147,·)$, $\chi_{440}(9,·)$, $\chi_{440}(267,·)$, $\chi_{440}(81,·)$, $\chi_{440}(3,·)$, $\chi_{440}(203,·)$, $\chi_{440}(89,·)$, $\chi_{440}(27,·)$, $\chi_{440}(361,·)$, $\chi_{440}(289,·)$, $\chi_{440}(163,·)$, $\chi_{440}(401,·)$, $\chi_{440}(169,·)$, $\chi_{440}(323,·)$, $\chi_{440}(427,·)$, $\chi_{440}(49,·)$, $\chi_{440}(243,·)$, $\chi_{440}(201,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{1054855908478} a^{18} + \frac{40274534656}{527427954239} a^{17} + \frac{1600657844}{527427954239} a^{16} + \frac{39130263459}{1054855908478} a^{15} - \frac{52882400951}{1054855908478} a^{14} + \frac{151606853083}{1054855908478} a^{13} - \frac{200202658757}{1054855908478} a^{12} + \frac{179937412937}{1054855908478} a^{11} - \frac{35799794964}{527427954239} a^{10} - \frac{78309698625}{527427954239} a^{9} - \frac{56678828105}{527427954239} a^{8} + \frac{16881612043}{1054855908478} a^{7} - \frac{414260889493}{1054855908478} a^{6} + \frac{234378754158}{527427954239} a^{5} + \frac{89001208760}{527427954239} a^{4} + \frac{234198840155}{1054855908478} a^{3} + \frac{124470694961}{527427954239} a^{2} - \frac{405419500363}{1054855908478} a - \frac{238256932417}{527427954239}$, $\frac{1}{4281612904016756014275834415257362} a^{19} + \frac{356839917976584766098}{2140806452008378007137917207628681} a^{18} - \frac{469404541576807839543254821777165}{4281612904016756014275834415257362} a^{17} - \frac{100669301795353582444284920539516}{2140806452008378007137917207628681} a^{16} - \frac{640735048709489303988157420342601}{4281612904016756014275834415257362} a^{15} - \frac{113443681673730772324109103754957}{2140806452008378007137917207628681} a^{14} - \frac{308307277071703406649013033044728}{2140806452008378007137917207628681} a^{13} - \frac{137731062696940094565080993329823}{2140806452008378007137917207628681} a^{12} - \frac{76827016288093736526983271039509}{2140806452008378007137917207628681} a^{11} - \frac{187694723175284698970262223131671}{2140806452008378007137917207628681} a^{10} + \frac{1734023741209125876601717251550599}{4281612904016756014275834415257362} a^{9} + \frac{903863481330193526154637629978171}{2140806452008378007137917207628681} a^{8} - \frac{473022716048570545887188109064747}{4281612904016756014275834415257362} a^{7} + \frac{538899403773827982126205094530121}{2140806452008378007137917207628681} a^{6} - \frac{702440456892451104117273287855873}{2140806452008378007137917207628681} a^{5} - \frac{119954300878522211778992169161992}{2140806452008378007137917207628681} a^{4} + \frac{266346427291372021630324421700195}{4281612904016756014275834415257362} a^{3} + \frac{652327367854083664825269847776434}{2140806452008378007137917207628681} a^{2} - \frac{2105075437810945802374006820763427}{4281612904016756014275834415257362} a - \frac{174835530679147942812379266343171}{2140806452008378007137917207628681}$

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

Class group and class number

$C_{5}$, which has order $5$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 107373870215 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.8000.1, \(\Q(\zeta_{11})^+\), 10.10.669871503125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
5Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$