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Properties

Label 20.8.23570143479...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 19^{2}\cdot 401^{8}$
Root discriminant $33.01$
Ramified primes $5, 19, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T141

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, -9, -74, -88, 161, 536, 309, 90, -50, -203, 244, -174, -122, 10, -168, 125, -67, 26, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 26*x^18 - 67*x^17 + 125*x^16 - 168*x^15 + 10*x^14 - 122*x^13 - 174*x^12 + 244*x^11 - 203*x^10 - 50*x^9 + 90*x^8 + 309*x^7 + 536*x^6 + 161*x^5 - 88*x^4 - 74*x^3 - 9*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 26*x^18 - 67*x^17 + 125*x^16 - 168*x^15 + 10*x^14 - 122*x^13 - 174*x^12 + 244*x^11 - 203*x^10 - 50*x^9 + 90*x^8 + 309*x^7 + 536*x^6 + 161*x^5 - 88*x^4 - 74*x^3 - 9*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 26 x^{18} - 67 x^{17} + 125 x^{16} - 168 x^{15} + 10 x^{14} - 122 x^{13} - 174 x^{12} + 244 x^{11} - 203 x^{10} - 50 x^{9} + 90 x^{8} + 309 x^{7} + 536 x^{6} + 161 x^{5} - 88 x^{4} - 74 x^{3} - 9 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2357014347930150805034775390625=5^{10}\cdot 19^{2}\cdot 401^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.01$
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:  $5, 19, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2} + \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{27} a^{15} - \frac{1}{27} a^{14} + \frac{2}{27} a^{12} + \frac{2}{27} a^{11} + \frac{4}{27} a^{10} + \frac{1}{9} a^{9} - \frac{2}{27} a^{8} + \frac{11}{27} a^{7} + \frac{1}{27} a^{6} + \frac{4}{9} a^{5} - \frac{8}{27} a^{4} - \frac{5}{27} a^{3} + \frac{5}{27} a^{2} + \frac{4}{9} a + \frac{8}{27}$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{14} - \frac{1}{27} a^{13} - \frac{2}{27} a^{12} + \frac{1}{9} a^{11} + \frac{4}{27} a^{10} + \frac{1}{27} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{13}{27} a^{6} - \frac{11}{27} a^{5} - \frac{7}{27} a^{4} + \frac{1}{9} a^{3} + \frac{11}{27} a^{2} + \frac{2}{27} a + \frac{5}{27}$, $\frac{1}{27} a^{17} + \frac{1}{27} a^{14} + \frac{1}{27} a^{13} + \frac{2}{27} a^{12} - \frac{1}{9} a^{11} + \frac{2}{27} a^{10} + \frac{1}{9} a^{9} + \frac{4}{27} a^{8} - \frac{4}{9} a^{7} + \frac{5}{27} a^{6} + \frac{11}{27} a^{5} - \frac{2}{27} a^{4} - \frac{1}{9} a^{3} - \frac{8}{27} a^{2} + \frac{11}{27} a + \frac{5}{27}$, $\frac{1}{243} a^{18} + \frac{1}{243} a^{17} - \frac{2}{243} a^{16} + \frac{4}{243} a^{15} + \frac{1}{243} a^{14} - \frac{10}{243} a^{13} + \frac{5}{81} a^{12} - \frac{7}{243} a^{11} + \frac{10}{81} a^{10} + \frac{23}{243} a^{9} - \frac{23}{243} a^{8} + \frac{56}{243} a^{7} - \frac{7}{243} a^{6} - \frac{116}{243} a^{5} + \frac{38}{81} a^{4} + \frac{1}{243} a^{3} + \frac{119}{243} a^{2} + \frac{1}{81} a + \frac{112}{243}$, $\frac{1}{3079232309743983} a^{19} + \frac{6055469246624}{3079232309743983} a^{18} + \frac{6061593308690}{3079232309743983} a^{17} - \frac{39187827280948}{3079232309743983} a^{16} + \frac{37352959422911}{3079232309743983} a^{15} - \frac{5668540889039}{1026410769914661} a^{14} + \frac{116910992310482}{3079232309743983} a^{13} + \frac{413544208593977}{3079232309743983} a^{12} + \frac{268498001494154}{3079232309743983} a^{11} + \frac{5701295488850}{3079232309743983} a^{10} + \frac{42115132612664}{1026410769914661} a^{9} + \frac{2727945563989}{114045641101629} a^{8} + \frac{1032388262302510}{3079232309743983} a^{7} + \frac{156879057008929}{342136923304887} a^{6} + \frac{1195315639904215}{3079232309743983} a^{5} - \frac{342170657683508}{3079232309743983} a^{4} + \frac{428638925686070}{1026410769914661} a^{3} - \frac{1058111193523741}{3079232309743983} a^{2} - \frac{577193148209819}{3079232309743983} a + \frac{262813303833289}{3079232309743983}$

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:  $13$
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:  \( 93484344.3404 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T141:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n141
Character table for t20n141 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.160801.1, 10.10.80803005003125.1, 10.4.491282270419.1, 10.4.1535257095059375.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed