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Properties

Label 20.4.20245771322...4048.4
Degree $20$
Signature $[4, 8]$
Discriminant $2^{30}\cdot 3^{18}\cdot 13^{5}\cdot 107^{4}$
Root discriminant $36.76$
Ramified primes $2, 3, 13, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![117, -540, 774, -1176, 2700, -5412, 8050, -7612, 4891, -3042, 3180, -3600, 2788, -1342, 544, -408, 330, -168, 52, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 52*x^18 - 168*x^17 + 330*x^16 - 408*x^15 + 544*x^14 - 1342*x^13 + 2788*x^12 - 3600*x^11 + 3180*x^10 - 3042*x^9 + 4891*x^8 - 7612*x^7 + 8050*x^6 - 5412*x^5 + 2700*x^4 - 1176*x^3 + 774*x^2 - 540*x + 117)
 
gp: K = bnfinit(x^20 - 10*x^19 + 52*x^18 - 168*x^17 + 330*x^16 - 408*x^15 + 544*x^14 - 1342*x^13 + 2788*x^12 - 3600*x^11 + 3180*x^10 - 3042*x^9 + 4891*x^8 - 7612*x^7 + 8050*x^6 - 5412*x^5 + 2700*x^4 - 1176*x^3 + 774*x^2 - 540*x + 117, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 52 x^{18} - 168 x^{17} + 330 x^{16} - 408 x^{15} + 544 x^{14} - 1342 x^{13} + 2788 x^{12} - 3600 x^{11} + 3180 x^{10} - 3042 x^{9} + 4891 x^{8} - 7612 x^{7} + 8050 x^{6} - 5412 x^{5} + 2700 x^{4} - 1176 x^{3} + 774 x^{2} - 540 x + 117 \)

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:  \(20245771322351194925844343554048=2^{30}\cdot 3^{18}\cdot 13^{5}\cdot 107^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.76$
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:  $2, 3, 13, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{6}$, $\frac{1}{27} a^{18} + \frac{1}{9} a^{15} + \frac{1}{9} a^{13} - \frac{2}{27} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{2}{9} a^{7} - \frac{8}{27} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{5763997560728895233398497} a^{19} + \frac{2932615956682712604299}{213481391138107230866611} a^{18} + \frac{7135150367768207559719}{1921332520242965077799499} a^{17} + \frac{212666086585898272892}{213481391138107230866611} a^{16} + \frac{21136492464624400600031}{213481391138107230866611} a^{15} - \frac{32880414227818876747513}{1921332520242965077799499} a^{14} + \frac{899686753573090621041595}{5763997560728895233398497} a^{13} - \frac{296759367982231609051346}{1921332520242965077799499} a^{12} + \frac{68697215559739737412991}{640444173414321692599833} a^{11} + \frac{299626462765633793750378}{1921332520242965077799499} a^{10} - \frac{57357796013026521724754}{640444173414321692599833} a^{9} - \frac{96074119130825384134475}{1921332520242965077799499} a^{8} - \frac{194667129996137487670835}{5763997560728895233398497} a^{7} + \frac{33816660454989511900160}{147794809249458852138423} a^{6} - \frac{206872455046524163681667}{1921332520242965077799499} a^{5} - \frac{660116965511804306338916}{1921332520242965077799499} a^{4} + \frac{78934396828311821684627}{213481391138107230866611} a^{3} - \frac{30908753775948557481785}{640444173414321692599833} a^{2} - \frac{75975579340040788799}{640444173414321692599833} a + \frac{8030223866872118477141}{16421645472162094682047}$

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:  \( 168093094.926 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.6.38998285028352.1

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

Sibling fields

Degree 20 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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$3$3.6.6.7$x^{6} + 3 x + 6$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
3.6.6.7$x^{6} + 3 x + 6$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.6.5.1$x^{6} - 52$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$107$$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
107.6.4.1$x^{6} + 1498 x^{3} + 1431125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$