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Properties

Label 20.2.11734933549...0000.1
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{30}\cdot 5^{10}\cdot 47^{9}$
Root discriminant $35.77$
Ramified primes $2, 5, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:S_4$ (as 20T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-169, -208, -1052, -348, 2194, 714, -914, 2410, 2431, -888, -1660, -92, -17, 6, 378, -62, -121, 40, 8, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 8*x^18 + 40*x^17 - 121*x^16 - 62*x^15 + 378*x^14 + 6*x^13 - 17*x^12 - 92*x^11 - 1660*x^10 - 888*x^9 + 2431*x^8 + 2410*x^7 - 914*x^6 + 714*x^5 + 2194*x^4 - 348*x^3 - 1052*x^2 - 208*x - 169)
 
gp: K = bnfinit(x^20 - 6*x^19 + 8*x^18 + 40*x^17 - 121*x^16 - 62*x^15 + 378*x^14 + 6*x^13 - 17*x^12 - 92*x^11 - 1660*x^10 - 888*x^9 + 2431*x^8 + 2410*x^7 - 914*x^6 + 714*x^5 + 2194*x^4 - 348*x^3 - 1052*x^2 - 208*x - 169, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 8 x^{18} + 40 x^{17} - 121 x^{16} - 62 x^{15} + 378 x^{14} + 6 x^{13} - 17 x^{12} - 92 x^{11} - 1660 x^{10} - 888 x^{9} + 2431 x^{8} + 2410 x^{7} - 914 x^{6} + 714 x^{5} + 2194 x^{4} - 348 x^{3} - 1052 x^{2} - 208 x - 169 \)

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:  \(-11734933549642070097920000000000=-\,2^{30}\cdot 5^{10}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.77$
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:  $2, 5, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{15} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{13} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{65} a^{18} - \frac{3}{65} a^{17} - \frac{1}{65} a^{16} + \frac{24}{65} a^{15} - \frac{23}{65} a^{14} + \frac{5}{13} a^{13} + \frac{24}{65} a^{12} + \frac{22}{65} a^{10} - \frac{2}{5} a^{9} + \frac{6}{13} a^{8} + \frac{8}{65} a^{7} + \frac{24}{65} a^{6} + \frac{12}{65} a^{5} - \frac{4}{13} a^{4} - \frac{9}{65} a^{3} + \frac{9}{65} a^{2} - \frac{9}{65} a - \frac{1}{5}$, $\frac{1}{4463548002524436436353806022213475} a^{19} - \frac{24046416712411139374011572682813}{4463548002524436436353806022213475} a^{18} + \frac{179093076976233822240763373868479}{4463548002524436436353806022213475} a^{17} + \frac{110434654440121614261284749329077}{4463548002524436436353806022213475} a^{16} - \frac{152695982142743203576380727346399}{892709600504887287270761204442695} a^{15} - \frac{288943287781762068567818451988382}{4463548002524436436353806022213475} a^{14} - \frac{97316890499310057804426441578568}{4463548002524436436353806022213475} a^{13} + \frac{1728313306089764126189943721641217}{4463548002524436436353806022213475} a^{12} + \frac{1954364193637008766388470461108039}{4463548002524436436353806022213475} a^{11} - \frac{230716349703219916981072247751724}{892709600504887287270761204442695} a^{10} + \frac{178986489373748671165360408675069}{892709600504887287270761204442695} a^{9} - \frac{2210519349551375737783047943340918}{4463548002524436436353806022213475} a^{8} - \frac{163101449101749375792151186119436}{343349846348033572027215847862575} a^{7} - \frac{1808107253053726591736584876020994}{4463548002524436436353806022213475} a^{6} - \frac{1650673383385119570757973129413201}{4463548002524436436353806022213475} a^{5} - \frac{582075736234923198648785635070049}{4463548002524436436353806022213475} a^{4} + \frac{235825513137748917429396440972567}{4463548002524436436353806022213475} a^{3} - \frac{1806031991428437088313786421021802}{4463548002524436436353806022213475} a^{2} + \frac{839436155954965160671811622494267}{4463548002524436436353806022213475} a + \frac{85186584783044157032130986688491}{343349846348033572027215847862575}$

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:  $10$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 42178724.6457 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:S_4$ (as 20T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 13 conjugacy class representatives for $C_5:S_4$
Character table for $C_5:S_4$

Intermediate fields

4.2.75200.1, 5.1.2209.1

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

Sibling fields

Degree 30 siblings: data not computed
Degree 40 sibling: 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 $15{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ R $15{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
47Data not computed