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Properties

Label 20.20.1406417496...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{8}\cdot 5^{10}\cdot 17^{8}\cdot 73^{8}$
Root discriminant $50.98$
Ramified primes $2, 5, 17, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29, 678, 4107, 6319, -13867, -45082, -4166, 89029, 58815, -67360, -71110, 16492, 31929, -278, -7051, -379, 825, 42, -48, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 48*x^18 + 42*x^17 + 825*x^16 - 379*x^15 - 7051*x^14 - 278*x^13 + 31929*x^12 + 16492*x^11 - 71110*x^10 - 67360*x^9 + 58815*x^8 + 89029*x^7 - 4166*x^6 - 45082*x^5 - 13867*x^4 + 6319*x^3 + 4107*x^2 + 678*x + 29)
 
gp: K = bnfinit(x^20 - x^19 - 48*x^18 + 42*x^17 + 825*x^16 - 379*x^15 - 7051*x^14 - 278*x^13 + 31929*x^12 + 16492*x^11 - 71110*x^10 - 67360*x^9 + 58815*x^8 + 89029*x^7 - 4166*x^6 - 45082*x^5 - 13867*x^4 + 6319*x^3 + 4107*x^2 + 678*x + 29, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 48 x^{18} + 42 x^{17} + 825 x^{16} - 379 x^{15} - 7051 x^{14} - 278 x^{13} + 31929 x^{12} + 16492 x^{11} - 71110 x^{10} - 67360 x^{9} + 58815 x^{8} + 89029 x^{7} - 4166 x^{6} - 45082 x^{5} - 13867 x^{4} + 6319 x^{3} + 4107 x^{2} + 678 x + 29 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14064174967249198299016802500000000=2^{8}\cdot 5^{10}\cdot 17^{8}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.98$
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:  $2, 5, 17, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{1176458198374841456457752636} a^{19} + \frac{118873829280245545007218605}{1176458198374841456457752636} a^{18} + \frac{26618509463521680715389141}{1176458198374841456457752636} a^{17} + \frac{10119940535337114698211173}{294114549593710364114438159} a^{16} - \frac{22747696628243821224864593}{294114549593710364114438159} a^{15} - \frac{11776346719775772559084263}{588229099187420728228876318} a^{14} + \frac{10488190056197003395553787}{588229099187420728228876318} a^{13} + \frac{41641734018309656372970417}{1176458198374841456457752636} a^{12} - \frac{111391982238782003550901122}{294114549593710364114438159} a^{11} + \frac{188303501035421169994940993}{588229099187420728228876318} a^{10} + \frac{157144444430389381007595907}{588229099187420728228876318} a^{9} - \frac{112988823941024372769586932}{294114549593710364114438159} a^{8} - \frac{331423361984536386461363793}{1176458198374841456457752636} a^{7} + \frac{536716714667531404882855561}{1176458198374841456457752636} a^{6} + \frac{48142678960561629081904593}{1176458198374841456457752636} a^{5} + \frac{54480472798095740007871036}{294114549593710364114438159} a^{4} + \frac{168528866877269185110543419}{588229099187420728228876318} a^{3} + \frac{132530022019086046583690572}{294114549593710364114438159} a^{2} + \frac{115548996384652751227575826}{294114549593710364114438159} a - \frac{334342886612434249897576487}{1176458198374841456457752636}$

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:  \( 104465887424 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1920
The 24 conjugacy class representatives for t20n230
Character table for t20n230 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.6160324.1, 10.10.4743698973122000.1, 10.10.118592474328050000.1, 10.10.948739794624400.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 32 sibling: 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$