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Properties

Label 20.0.22392111040...9121.2
Degree $20$
Signature $[0, 10]$
Discriminant $17^{4}\cdot 401^{9}$
Root discriminant $26.15$
Ramified primes $17, 401$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T87)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -432, 774, -462, -233, 617, -44, -597, 1005, -636, -170, 278, -31, 59, -98, 0, 42, -12, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 2*x^18 - 12*x^17 + 42*x^16 - 98*x^14 + 59*x^13 - 31*x^12 + 278*x^11 - 170*x^10 - 636*x^9 + 1005*x^8 - 597*x^7 - 44*x^6 + 617*x^5 - 233*x^4 - 462*x^3 + 774*x^2 - 432*x + 81)
 
gp: K = bnfinit(x^20 - x^19 - 2*x^18 - 12*x^17 + 42*x^16 - 98*x^14 + 59*x^13 - 31*x^12 + 278*x^11 - 170*x^10 - 636*x^9 + 1005*x^8 - 597*x^7 - 44*x^6 + 617*x^5 - 233*x^4 - 462*x^3 + 774*x^2 - 432*x + 81, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 2 x^{18} - 12 x^{17} + 42 x^{16} - 98 x^{14} + 59 x^{13} - 31 x^{12} + 278 x^{11} - 170 x^{10} - 636 x^{9} + 1005 x^{8} - 597 x^{7} - 44 x^{6} + 617 x^{5} - 233 x^{4} - 462 x^{3} + 774 x^{2} - 432 x + 81 \)

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:  \(22392111040671733867877719121=17^{4}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.15$
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:  $17, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{15} + \frac{1}{18} a^{14} - \frac{1}{18} a^{13} + \frac{1}{9} a^{12} - \frac{5}{18} a^{11} + \frac{1}{18} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{4}{9} a^{7} + \frac{7}{18} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{16} - \frac{1}{9} a^{14} - \frac{1}{6} a^{13} - \frac{1}{18} a^{12} - \frac{1}{3} a^{11} + \frac{4}{9} a^{10} + \frac{1}{6} a^{9} - \frac{1}{9} a^{8} - \frac{1}{6} a^{7} - \frac{5}{18} a^{6} - \frac{2}{9} a^{4} - \frac{1}{3} a^{3} - \frac{7}{18} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{18} a^{17} - \frac{1}{18} a^{14} - \frac{1}{6} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{5}{18} a^{10} + \frac{2}{9} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{2}{9} a^{6} - \frac{4}{9} a^{4} + \frac{7}{18} a^{3} + \frac{1}{9} a^{2} - \frac{1}{2} a$, $\frac{1}{234} a^{18} + \frac{5}{234} a^{17} - \frac{1}{117} a^{16} - \frac{5}{234} a^{15} - \frac{16}{117} a^{14} + \frac{23}{234} a^{13} + \frac{2}{39} a^{12} + \frac{1}{6} a^{11} - \frac{3}{26} a^{10} + \frac{77}{234} a^{9} - \frac{34}{117} a^{8} + \frac{7}{18} a^{7} - \frac{31}{117} a^{6} + \frac{16}{117} a^{5} + \frac{5}{26} a^{4} - \frac{7}{78} a^{3} - \frac{19}{234} a^{2} + \frac{31}{78} a + \frac{6}{13}$, $\frac{1}{306667932688982487072294} a^{19} + \frac{102919691157377253881}{306667932688982487072294} a^{18} + \frac{3128295239047967746073}{153333966344491243536147} a^{17} - \frac{346593818850293153675}{102222644229660829024098} a^{16} + \frac{2678470733411467695011}{102222644229660829024098} a^{15} + \frac{1882101762302577594230}{17037107371610138170683} a^{14} + \frac{27596613570902369186095}{306667932688982487072294} a^{13} + \frac{19616916852912739047959}{306667932688982487072294} a^{12} - \frac{20068506284053567452314}{153333966344491243536147} a^{11} - \frac{31236708494913833734532}{153333966344491243536147} a^{10} - \frac{105539351240042786025419}{306667932688982487072294} a^{9} - \frac{8604275120263260870827}{34074214743220276341366} a^{8} - \frac{542366447973776575649}{51111322114830414512049} a^{7} + \frac{34683341515860390521627}{102222644229660829024098} a^{6} + \frac{131693790782337873678217}{306667932688982487072294} a^{5} - \frac{14427136389794444894275}{306667932688982487072294} a^{4} + \frac{70445309075864894104015}{306667932688982487072294} a^{3} - \frac{33776066806569467359355}{102222644229660829024098} a^{2} + \frac{8873775179717328221131}{34074214743220276341366} a + \frac{2126273664498799773901}{11358071581073425447122}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 870300.018994 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T87):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.6.7472661902689.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed