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Properties

Label 20.20.3825159923...5152.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{40}\cdot 17^{13}\cdot 37^{8}$
Root discriminant $106.94$
Ramified primes $2, 17, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2:F_5$ (as 20T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-263, -2960, 106752, -348424, -56571, 1143416, -463526, -1245448, 629427, 669932, -339336, -202968, 94810, 35964, -14718, -3644, 1265, 192, -56, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 56*x^18 + 192*x^17 + 1265*x^16 - 3644*x^15 - 14718*x^14 + 35964*x^13 + 94810*x^12 - 202968*x^11 - 339336*x^10 + 669932*x^9 + 629427*x^8 - 1245448*x^7 - 463526*x^6 + 1143416*x^5 - 56571*x^4 - 348424*x^3 + 106752*x^2 - 2960*x - 263)
 
gp: K = bnfinit(x^20 - 4*x^19 - 56*x^18 + 192*x^17 + 1265*x^16 - 3644*x^15 - 14718*x^14 + 35964*x^13 + 94810*x^12 - 202968*x^11 - 339336*x^10 + 669932*x^9 + 629427*x^8 - 1245448*x^7 - 463526*x^6 + 1143416*x^5 - 56571*x^4 - 348424*x^3 + 106752*x^2 - 2960*x - 263, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 56 x^{18} + 192 x^{17} + 1265 x^{16} - 3644 x^{15} - 14718 x^{14} + 35964 x^{13} + 94810 x^{12} - 202968 x^{11} - 339336 x^{10} + 669932 x^{9} + 629427 x^{8} - 1245448 x^{7} - 463526 x^{6} + 1143416 x^{5} - 56571 x^{4} - 348424 x^{3} + 106752 x^{2} - 2960 x - 263 \)

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:  \(38251599236941169123840379965550626865152=2^{40}\cdot 17^{13}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.94$
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:  $2, 17, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17} a^{16} - \frac{3}{17} a^{15} - \frac{2}{17} a^{14} + \frac{1}{17} a^{13} - \frac{2}{17} a^{12} + \frac{3}{17} a^{11} - \frac{4}{17} a^{10} + \frac{7}{17} a^{9} + \frac{4}{17} a^{7} + \frac{2}{17} a^{6} - \frac{2}{17} a^{5} + \frac{8}{17} a^{4} - \frac{4}{17} a^{3} + \frac{6}{17} a^{2} + \frac{6}{17} a - \frac{8}{17}$, $\frac{1}{17} a^{17} + \frac{6}{17} a^{15} - \frac{5}{17} a^{14} + \frac{1}{17} a^{13} - \frac{3}{17} a^{12} + \frac{5}{17} a^{11} - \frac{5}{17} a^{10} + \frac{4}{17} a^{9} + \frac{4}{17} a^{8} - \frac{3}{17} a^{7} + \frac{4}{17} a^{6} + \frac{2}{17} a^{5} + \frac{3}{17} a^{4} - \frac{6}{17} a^{3} + \frac{7}{17} a^{2} - \frac{7}{17} a - \frac{7}{17}$, $\frac{1}{2703} a^{18} - \frac{59}{2703} a^{17} + \frac{31}{2703} a^{16} + \frac{388}{901} a^{15} - \frac{445}{901} a^{14} + \frac{1102}{2703} a^{13} - \frac{871}{2703} a^{12} - \frac{1279}{2703} a^{11} - \frac{923}{2703} a^{10} + \frac{283}{2703} a^{9} + \frac{11}{901} a^{8} + \frac{326}{901} a^{7} - \frac{322}{901} a^{6} - \frac{403}{2703} a^{5} + \frac{70}{159} a^{4} - \frac{334}{2703} a^{3} + \frac{233}{901} a^{2} - \frac{56}{2703} a + \frac{77}{2703}$, $\frac{1}{380957951778538167036671442499896565017} a^{19} + \frac{37838049542735380326625615486327562}{380957951778538167036671442499896565017} a^{18} - \frac{7736820093923437479242331786582242458}{380957951778538167036671442499896565017} a^{17} + \frac{9951695523469691500923362230971366511}{380957951778538167036671442499896565017} a^{16} - \frac{33788786828283959776996714620601796582}{126985983926179389012223814166632188339} a^{15} + \frac{101255652548382389318040304782478110187}{380957951778538167036671442499896565017} a^{14} - \frac{3681311867110835963280672479370230672}{7469763760363493471307283186272481667} a^{13} - \frac{100345366571369011831787645434661438552}{380957951778538167036671442499896565017} a^{12} + \frac{30836739245643332635467973971837214498}{126985983926179389012223814166632188339} a^{11} + \frac{146192246573181449372451126229608353546}{380957951778538167036671442499896565017} a^{10} + \frac{10226068163299558609383337717615929535}{29304457829118320541282418653838197309} a^{9} - \frac{7453571291967406669203343546543164691}{126985983926179389012223814166632188339} a^{8} - \frac{63046338280727225967028830465987435416}{126985983926179389012223814166632188339} a^{7} + \frac{175665028974607075893143537175752339837}{380957951778538167036671442499896565017} a^{6} - \frac{117041419383307849309094495930757984779}{380957951778538167036671442499896565017} a^{5} + \frac{6015151633016478377910778908268308940}{380957951778538167036671442499896565017} a^{4} + \frac{110265271991300856978594035721283284755}{380957951778538167036671442499896565017} a^{3} - \frac{167996050693765751553505814949468066566}{380957951778538167036671442499896565017} a^{2} + \frac{1667855646226159139707098429145767583}{7469763760363493471307283186272481667} a - \frac{131037588456153889841336060724993074794}{380957951778538167036671442499896565017}$

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:  \( 110095846393000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.4352.1, 5.5.6725897.1, 10.10.1482348640816627712.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$