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Properties

Label 20.8.38627487549...3125.1
Degree $20$
Signature $[8, 6]$
Discriminant $3^{4}\cdot 5^{15}\cdot 23^{4}\cdot 89^{5}$
Root discriminant $23.95$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T887

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, -8, -84, 107, 86, 254, -1347, 1570, -416, -495, 624, -575, 342, -23, -84, 73, -46, 18, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 18*x^18 - 46*x^17 + 73*x^16 - 84*x^15 - 23*x^14 + 342*x^13 - 575*x^12 + 624*x^11 - 495*x^10 - 416*x^9 + 1570*x^8 - 1347*x^7 + 254*x^6 + 86*x^5 + 107*x^4 - 84*x^3 - 8*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^20 - 5*x^19 + 18*x^18 - 46*x^17 + 73*x^16 - 84*x^15 - 23*x^14 + 342*x^13 - 575*x^12 + 624*x^11 - 495*x^10 - 416*x^9 + 1570*x^8 - 1347*x^7 + 254*x^6 + 86*x^5 + 107*x^4 - 84*x^3 - 8*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 18 x^{18} - 46 x^{17} + 73 x^{16} - 84 x^{15} - 23 x^{14} + 342 x^{13} - 575 x^{12} + 624 x^{11} - 495 x^{10} - 416 x^{9} + 1570 x^{8} - 1347 x^{7} + 254 x^{6} + 86 x^{5} + 107 x^{4} - 84 x^{3} - 8 x^{2} + 8 x + 1 \)

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:  \(3862748754933970001220703125=3^{4}\cdot 5^{15}\cdot 23^{4}\cdot 89^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.95$
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:  $3, 5, 23, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{15} - \frac{1}{5} a^{14} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{15} + \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{15} a^{18} - \frac{1}{15} a^{17} + \frac{1}{15} a^{16} + \frac{7}{15} a^{15} + \frac{7}{15} a^{14} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{3} a^{10} + \frac{4}{15} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{3} a^{6} + \frac{4}{15} a^{5} + \frac{2}{15} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{15}$, $\frac{1}{122618831191245} a^{19} - \frac{6939235085}{2724862915361} a^{18} - \frac{3025224956317}{40872943730415} a^{17} - \frac{11647968050686}{122618831191245} a^{16} + \frac{19398929818037}{122618831191245} a^{15} + \frac{13535957126944}{122618831191245} a^{14} + \frac{1516886346116}{40872943730415} a^{13} - \frac{16845382673756}{40872943730415} a^{12} + \frac{37982251053232}{122618831191245} a^{11} + \frac{48028844818478}{122618831191245} a^{10} - \frac{8111110919332}{24523766238249} a^{9} - \frac{13927119176123}{40872943730415} a^{8} - \frac{7516525927013}{122618831191245} a^{7} + \frac{52487445096692}{122618831191245} a^{6} - \frac{3756999402613}{8174588746083} a^{5} - \frac{36214836988708}{122618831191245} a^{4} - \frac{16387538112154}{40872943730415} a^{3} - \frac{708531736704}{13624314576805} a^{2} - \frac{9519673620986}{122618831191245} a - \frac{24924391602431}{122618831191245}$

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:  \( 2054207.1748 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T887:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 245760
The 201 conjugacy class representatives for t20n887 are not computed
Character table for t20n887 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.767625.1, 10.10.2946240703125.1

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 $20$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
89Data not computed