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Properties

Label 20.0.51109769008...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 13^{4}\cdot 41^{7}\cdot 97^{2}$
Root discriminant $21.65$
Ramified primes $5, 13, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8159, -27921, 50935, -63953, 63769, -49129, 24457, -2629, -6521, 5811, -2581, 362, 579, -565, 192, 26, -42, 16, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 3*x^18 + 16*x^17 - 42*x^16 + 26*x^15 + 192*x^14 - 565*x^13 + 579*x^12 + 362*x^11 - 2581*x^10 + 5811*x^9 - 6521*x^8 - 2629*x^7 + 24457*x^6 - 49129*x^5 + 63769*x^4 - 63953*x^3 + 50935*x^2 - 27921*x + 8159)
 
gp: K = bnfinit(x^20 - 2*x^19 - 3*x^18 + 16*x^17 - 42*x^16 + 26*x^15 + 192*x^14 - 565*x^13 + 579*x^12 + 362*x^11 - 2581*x^10 + 5811*x^9 - 6521*x^8 - 2629*x^7 + 24457*x^6 - 49129*x^5 + 63769*x^4 - 63953*x^3 + 50935*x^2 - 27921*x + 8159, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 3 x^{18} + 16 x^{17} - 42 x^{16} + 26 x^{15} + 192 x^{14} - 565 x^{13} + 579 x^{12} + 362 x^{11} - 2581 x^{10} + 5811 x^{9} - 6521 x^{8} - 2629 x^{7} + 24457 x^{6} - 49129 x^{5} + 63769 x^{4} - 63953 x^{3} + 50935 x^{2} - 27921 x + 8159 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(511097690085059595400390625=5^{10}\cdot 13^{4}\cdot 41^{7}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.65$
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:  $5, 13, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23} a^{18} - \frac{7}{23} a^{17} - \frac{4}{23} a^{16} - \frac{11}{23} a^{15} - \frac{4}{23} a^{14} + \frac{5}{23} a^{13} - \frac{11}{23} a^{12} + \frac{9}{23} a^{10} - \frac{5}{23} a^{9} - \frac{5}{23} a^{8} - \frac{10}{23} a^{7} + \frac{11}{23} a^{6} - \frac{1}{23} a^{5} + \frac{8}{23} a^{4} - \frac{5}{23} a^{3} + \frac{3}{23} a^{2} - \frac{9}{23} a - \frac{4}{23}$, $\frac{1}{707700481150664827788171927077372988763} a^{19} + \frac{13851629638291239770553394228733234978}{707700481150664827788171927077372988763} a^{18} + \frac{347948190863596856148992434849506140972}{707700481150664827788171927077372988763} a^{17} - \frac{165907608903731598353020957413188590314}{707700481150664827788171927077372988763} a^{16} - \frac{105567982532282307902031106552649809595}{707700481150664827788171927077372988763} a^{15} + \frac{257607899587865292749400309541976813885}{707700481150664827788171927077372988763} a^{14} - \frac{325375706643121084028640998190131550387}{707700481150664827788171927077372988763} a^{13} - \frac{124598304871525328654840055513245015096}{707700481150664827788171927077372988763} a^{12} + \frac{123344057039196995915887610668071362755}{707700481150664827788171927077372988763} a^{11} + \frac{14860128072975918893527674992658090695}{30769586136985427295137909872929260381} a^{10} - \frac{77598507003321310842337186327793841224}{707700481150664827788171927077372988763} a^{9} + \frac{6475158513517945540951598762063370600}{30769586136985427295137909872929260381} a^{8} + \frac{309261378655154666054537402323242098398}{707700481150664827788171927077372988763} a^{7} - \frac{6394959692715629241690081114621943646}{30769586136985427295137909872929260381} a^{6} + \frac{136913396162519872729702831120662424152}{707700481150664827788171927077372988763} a^{5} - \frac{202404261246614799533894286587552992368}{707700481150664827788171927077372988763} a^{4} - \frac{160402875380295532448219792156029379825}{707700481150664827788171927077372988763} a^{3} + \frac{221465240941086648318748828899545746096}{707700481150664827788171927077372988763} a^{2} - \frac{89351867893142862837691134024509513803}{707700481150664827788171927077372988763} a - \frac{313244427853583699041609919993047472559}{707700481150664827788171927077372988763}$

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:  $9$
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:  \( 95787.007316 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.2665.1, 10.2.887778125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ R $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.12.0.1$x^{12} + x^{2} - x + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$
41Data not computed
$97$97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$
97.8.0.1$x^{8} - x + 84$$1$$8$$0$$C_8$$[\ ]^{8}$