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Properties

Label 20.20.1315275659...8673.1
Degree $20$
Signature $[20, 0]$
Discriminant $11^{16}\cdot 17^{15}$
Root discriminant $57.01$
Ramified primes $11, 17$
Class number $1$ (GRH)
Class group Trivial (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![1033, -2449, -20309, 17397, 107725, -31515, -226443, 22455, 237299, -4409, -137522, -2629, 45828, 1660, -8765, -354, 926, 32, -49, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 49*x^18 + 32*x^17 + 926*x^16 - 354*x^15 - 8765*x^14 + 1660*x^13 + 45828*x^12 - 2629*x^11 - 137522*x^10 - 4409*x^9 + 237299*x^8 + 22455*x^7 - 226443*x^6 - 31515*x^5 + 107725*x^4 + 17397*x^3 - 20309*x^2 - 2449*x + 1033)
 
gp: K = bnfinit(x^20 - x^19 - 49*x^18 + 32*x^17 + 926*x^16 - 354*x^15 - 8765*x^14 + 1660*x^13 + 45828*x^12 - 2629*x^11 - 137522*x^10 - 4409*x^9 + 237299*x^8 + 22455*x^7 - 226443*x^6 - 31515*x^5 + 107725*x^4 + 17397*x^3 - 20309*x^2 - 2449*x + 1033, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 49 x^{18} + 32 x^{17} + 926 x^{16} - 354 x^{15} - 8765 x^{14} + 1660 x^{13} + 45828 x^{12} - 2629 x^{11} - 137522 x^{10} - 4409 x^{9} + 237299 x^{8} + 22455 x^{7} - 226443 x^{6} - 31515 x^{5} + 107725 x^{4} + 17397 x^{3} - 20309 x^{2} - 2449 x + 1033 \)

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:  \(131527565972137936816816034072938673=11^{16}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.01$
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:  $11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(187=11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{187}(64,·)$, $\chi_{187}(1,·)$, $\chi_{187}(67,·)$, $\chi_{187}(4,·)$, $\chi_{187}(69,·)$, $\chi_{187}(135,·)$, $\chi_{187}(137,·)$, $\chi_{187}(16,·)$, $\chi_{187}(81,·)$, $\chi_{187}(86,·)$, $\chi_{187}(152,·)$, $\chi_{187}(89,·)$, $\chi_{187}(157,·)$, $\chi_{187}(166,·)$, $\chi_{187}(38,·)$, $\chi_{187}(103,·)$, $\chi_{187}(169,·)$, $\chi_{187}(174,·)$, $\chi_{187}(47,·)$, $\chi_{187}(115,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2258404151618334475175002761674878666} a^{19} - \frac{394340400029153125003652821433137061}{2258404151618334475175002761674878666} a^{18} + \frac{271115295882140560145852905333391643}{1129202075809167237587501380837439333} a^{17} - \frac{58944976980178268156131112961360329}{2258404151618334475175002761674878666} a^{16} - \frac{372941101042152394645071215326327255}{2258404151618334475175002761674878666} a^{15} + \frac{340918407545081289591926710622811868}{1129202075809167237587501380837439333} a^{14} - \frac{582379291007296373142554876600786769}{2258404151618334475175002761674878666} a^{13} + \frac{886049215457833639494168064273616433}{2258404151618334475175002761674878666} a^{12} - \frac{118098448248767134109231410020462859}{1129202075809167237587501380837439333} a^{11} + \frac{69996040050637711302402805550680471}{1129202075809167237587501380837439333} a^{10} - \frac{572316135132531383541158484466580267}{2258404151618334475175002761674878666} a^{9} + \frac{627599029026407744551586187116000801}{2258404151618334475175002761674878666} a^{8} + \frac{21386151332601085045808965294856367}{2258404151618334475175002761674878666} a^{7} + \frac{349523396365557291934265788030263433}{2258404151618334475175002761674878666} a^{6} + \frac{100192220350297268561590824006492131}{1129202075809167237587501380837439333} a^{5} - \frac{898208561817034047007089651904218179}{2258404151618334475175002761674878666} a^{4} - \frac{900882311288978464613940466572055103}{2258404151618334475175002761674878666} a^{3} + \frac{322643787298949251775678943071331007}{1129202075809167237587501380837439333} a^{2} + \frac{456860064408751177664471591860454289}{2258404151618334475175002761674878666} a - \frac{204387236089121975349946130474098813}{1129202075809167237587501380837439333}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 124763101953 \) (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

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{11})^+\), 10.10.304358957700017.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ $20$ $20$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
11Data not computed
17Data not computed