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Properties

Label 20.8.38282048095...0625.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{10}\cdot 19^{6}\cdot 1699^{4}$
Root discriminant $23.94$
Ramified primes $5, 19, 1699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T760

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, 48, 322, 452, 306, 172, -1, -540, 44, -51, -260, 650, -574, 457, -316, 123, -37, 13, 4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 4*x^18 + 13*x^17 - 37*x^16 + 123*x^15 - 316*x^14 + 457*x^13 - 574*x^12 + 650*x^11 - 260*x^10 - 51*x^9 + 44*x^8 - 540*x^7 - x^6 + 172*x^5 + 306*x^4 + 452*x^3 + 322*x^2 + 48*x - 29)
 
gp: K = bnfinit(x^20 - 5*x^19 + 4*x^18 + 13*x^17 - 37*x^16 + 123*x^15 - 316*x^14 + 457*x^13 - 574*x^12 + 650*x^11 - 260*x^10 - 51*x^9 + 44*x^8 - 540*x^7 - x^6 + 172*x^5 + 306*x^4 + 452*x^3 + 322*x^2 + 48*x - 29, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 4 x^{18} + 13 x^{17} - 37 x^{16} + 123 x^{15} - 316 x^{14} + 457 x^{13} - 574 x^{12} + 650 x^{11} - 260 x^{10} - 51 x^{9} + 44 x^{8} - 540 x^{7} - x^{6} + 172 x^{5} + 306 x^{4} + 452 x^{3} + 322 x^{2} + 48 x - 29 \)

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:  \(3828204809593746045712890625=5^{10}\cdot 19^{6}\cdot 1699^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.94$
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:  $5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{25} a^{14} + \frac{2}{25} a^{13} - \frac{2}{25} a^{12} - \frac{1}{25} a^{11} - \frac{4}{25} a^{10} + \frac{12}{25} a^{9} + \frac{3}{25} a^{8} + \frac{6}{25} a^{7} - \frac{1}{25} a^{6} - \frac{1}{25} a^{5} + \frac{3}{25} a^{4} - \frac{12}{25} a^{3} - \frac{1}{5} a^{2} - \frac{6}{25} a - \frac{11}{25}$, $\frac{1}{25} a^{15} - \frac{1}{25} a^{13} - \frac{2}{25} a^{12} + \frac{3}{25} a^{11} - \frac{6}{25} a^{9} - \frac{2}{5} a^{8} + \frac{7}{25} a^{7} - \frac{4}{25} a^{6} - \frac{2}{5} a^{5} - \frac{3}{25} a^{4} - \frac{11}{25} a^{3} + \frac{9}{25} a^{2} + \frac{11}{25} a + \frac{12}{25}$, $\frac{1}{125} a^{16} - \frac{2}{125} a^{15} - \frac{1}{125} a^{14} + \frac{7}{125} a^{12} - \frac{31}{125} a^{11} - \frac{56}{125} a^{10} - \frac{48}{125} a^{9} + \frac{52}{125} a^{8} + \frac{57}{125} a^{7} - \frac{27}{125} a^{6} - \frac{8}{125} a^{5} + \frac{9}{25} a^{4} - \frac{44}{125} a^{3} - \frac{57}{125} a^{2} - \frac{2}{25} a + \frac{1}{125}$, $\frac{1}{125} a^{17} - \frac{2}{125} a^{14} + \frac{2}{125} a^{13} - \frac{2}{125} a^{12} + \frac{47}{125} a^{11} + \frac{8}{25} a^{10} - \frac{24}{125} a^{9} + \frac{36}{125} a^{8} + \frac{47}{125} a^{7} - \frac{7}{125} a^{6} - \frac{21}{125} a^{5} - \frac{44}{125} a^{4} - \frac{1}{5} a^{3} - \frac{4}{125} a^{2} - \frac{39}{125} a - \frac{38}{125}$, $\frac{1}{625} a^{18} - \frac{1}{625} a^{17} + \frac{1}{625} a^{16} - \frac{9}{625} a^{15} + \frac{3}{625} a^{14} - \frac{24}{625} a^{13} - \frac{59}{625} a^{12} + \frac{272}{625} a^{11} + \frac{56}{125} a^{10} + \frac{42}{625} a^{9} - \frac{262}{625} a^{8} - \frac{182}{625} a^{7} + \frac{54}{625} a^{6} - \frac{156}{625} a^{5} - \frac{296}{625} a^{4} + \frac{257}{625} a^{3} - \frac{237}{625} a^{2} + \frac{86}{625} a + \frac{4}{625}$, $\frac{1}{260332922744375} a^{19} + \frac{10592026748}{52066584548875} a^{18} + \frac{151358830326}{52066584548875} a^{17} + \frac{107824161812}{260332922744375} a^{16} + \frac{1388170243524}{260332922744375} a^{15} + \frac{634621606054}{260332922744375} a^{14} - \frac{14602133690103}{260332922744375} a^{13} - \frac{1137425990879}{20025609441875} a^{12} - \frac{63569785961008}{260332922744375} a^{11} + \frac{36009251475017}{260332922744375} a^{10} - \frac{18607229882647}{52066584548875} a^{9} - \frac{113939520050444}{260332922744375} a^{8} - \frac{74508206673483}{260332922744375} a^{7} - \frac{50468026722967}{260332922744375} a^{6} + \frac{645919573959}{2683844564375} a^{5} - \frac{5252408593034}{260332922744375} a^{4} - \frac{3789001598754}{52066584548875} a^{3} + \frac{69769733307454}{260332922744375} a^{2} - \frac{5125245474868}{10413316909775} a - \frac{101368187356491}{260332922744375}$

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:  \( 1791091.8411 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T760:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n760 are not computed
Character table for t20n760 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

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
$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$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$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}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1699Data not computed