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Properties

Label 20.4.41327071510...0625.1
Degree $20$
Signature $[4, 8]$
Discriminant $3^{6}\cdot 5^{14}\cdot 23^{6}\cdot 89^{4}$
Root discriminant $26.97$
Ramified primes $3, 5, 23, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T794

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79, 281, 418, 52, -320, -312, 650, 650, -430, -1030, 283, 672, -179, -286, -25, 99, 47, -14, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 11*x^18 - 14*x^17 + 47*x^16 + 99*x^15 - 25*x^14 - 286*x^13 - 179*x^12 + 672*x^11 + 283*x^10 - 1030*x^9 - 430*x^8 + 650*x^7 + 650*x^6 - 312*x^5 - 320*x^4 + 52*x^3 + 418*x^2 + 281*x + 79)
 
gp: K = bnfinit(x^20 - 11*x^18 - 14*x^17 + 47*x^16 + 99*x^15 - 25*x^14 - 286*x^13 - 179*x^12 + 672*x^11 + 283*x^10 - 1030*x^9 - 430*x^8 + 650*x^7 + 650*x^6 - 312*x^5 - 320*x^4 + 52*x^3 + 418*x^2 + 281*x + 79, 1)
 

Normalized defining polynomial

\( x^{20} - 11 x^{18} - 14 x^{17} + 47 x^{16} + 99 x^{15} - 25 x^{14} - 286 x^{13} - 179 x^{12} + 672 x^{11} + 283 x^{10} - 1030 x^{9} - 430 x^{8} + 650 x^{7} + 650 x^{6} - 312 x^{5} - 320 x^{4} + 52 x^{3} + 418 x^{2} + 281 x + 79 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(41327071510653103765869140625=3^{6}\cdot 5^{14}\cdot 23^{6}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5}$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{15} - \frac{1}{25} a^{14} + \frac{2}{25} a^{13} + \frac{6}{25} a^{12} - \frac{7}{25} a^{11} - \frac{2}{5} a^{10} - \frac{7}{25} a^{9} + \frac{9}{25} a^{8} + \frac{2}{25} a^{7} + \frac{4}{25} a^{6} - \frac{8}{25} a^{5} + \frac{8}{25} a^{4} + \frac{4}{25} a^{3} + \frac{2}{25} a^{2} + \frac{12}{25} a + \frac{9}{25}$, $\frac{1}{25} a^{17} - \frac{2}{25} a^{15} + \frac{1}{25} a^{14} - \frac{2}{25} a^{13} + \frac{4}{25} a^{12} - \frac{12}{25} a^{11} + \frac{3}{25} a^{10} - \frac{8}{25} a^{9} - \frac{4}{25} a^{8} + \frac{11}{25} a^{7} - \frac{9}{25} a^{6} - \frac{2}{5} a^{5} + \frac{7}{25} a^{4} + \frac{1}{25} a^{3} + \frac{9}{25} a^{2} + \frac{11}{25} a - \frac{1}{25}$, $\frac{1}{625} a^{18} - \frac{3}{625} a^{17} - \frac{8}{625} a^{16} + \frac{28}{625} a^{15} + \frac{11}{625} a^{14} + \frac{23}{625} a^{13} - \frac{32}{125} a^{12} - \frac{69}{625} a^{11} - \frac{137}{625} a^{10} - \frac{128}{625} a^{9} + \frac{239}{625} a^{8} + \frac{271}{625} a^{7} + \frac{198}{625} a^{6} - \frac{14}{125} a^{5} + \frac{172}{625} a^{4} - \frac{33}{625} a^{3} - \frac{3}{625} a^{2} + \frac{9}{625} a + \frac{34}{625}$, $\frac{1}{124703340293541165460625} a^{19} + \frac{19297750805259057202}{124703340293541165460625} a^{18} - \frac{2089740305958481130548}{124703340293541165460625} a^{17} - \frac{15023079887160812812}{124703340293541165460625} a^{16} + \frac{2571723226771721909051}{124703340293541165460625} a^{15} - \frac{4807145573679503713447}{124703340293541165460625} a^{14} + \frac{614909964384213676251}{24940668058708233092125} a^{13} + \frac{45442349677786873266231}{124703340293541165460625} a^{12} - \frac{264941420982256549029}{1502449883054712836875} a^{11} + \frac{4895741302603063271762}{124703340293541165460625} a^{10} - \frac{11794509378051417575951}{124703340293541165460625} a^{9} + \frac{33593632853179574421666}{124703340293541165460625} a^{8} + \frac{42864935556468852322928}{124703340293541165460625} a^{7} - \frac{11028046455649404569766}{24940668058708233092125} a^{6} + \frac{51327550284118896434797}{124703340293541165460625} a^{5} - \frac{813550296700864617651}{5421884360588746324375} a^{4} + \frac{33816193394164219581632}{124703340293541165460625} a^{3} - \frac{19438551276377448821331}{124703340293541165460625} a^{2} + \frac{51366937681554518522804}{124703340293541165460625} a + \frac{7049326926347085492369}{24940668058708233092125}$

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:  $11$
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:  \( 2926683.55395 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T794:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 138 conjugacy class representatives for t20n794 are not computed
Character table for t20n794 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
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