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Properties

Label 20.4.32473988377...0464.5
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{17}\cdot 89^{4}$
Root discriminant $26.64$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-947, -4199, -5247, 776, 4605, 1788, 1497, -324, -4065, 517, 3455, -795, -1558, 405, 474, -170, -75, 34, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 7*x^18 + 34*x^17 - 75*x^16 - 170*x^15 + 474*x^14 + 405*x^13 - 1558*x^12 - 795*x^11 + 3455*x^10 + 517*x^9 - 4065*x^8 - 324*x^7 + 1497*x^6 + 1788*x^5 + 4605*x^4 + 776*x^3 - 5247*x^2 - 4199*x - 947)
 
gp: K = bnfinit(x^20 - 6*x^19 + 7*x^18 + 34*x^17 - 75*x^16 - 170*x^15 + 474*x^14 + 405*x^13 - 1558*x^12 - 795*x^11 + 3455*x^10 + 517*x^9 - 4065*x^8 - 324*x^7 + 1497*x^6 + 1788*x^5 + 4605*x^4 + 776*x^3 - 5247*x^2 - 4199*x - 947, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 7 x^{18} + 34 x^{17} - 75 x^{16} - 170 x^{15} + 474 x^{14} + 405 x^{13} - 1558 x^{12} - 795 x^{11} + 3455 x^{10} + 517 x^{9} - 4065 x^{8} - 324 x^{7} + 1497 x^{6} + 1788 x^{5} + 4605 x^{4} + 776 x^{3} - 5247 x^{2} - 4199 x - 947 \)

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:  \(32473988377432635504517950464=2^{10}\cdot 11^{17}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.64$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{2}{11} a^{15} - \frac{5}{11} a^{14} - \frac{3}{11} a^{13} + \frac{2}{11} a^{12} + \frac{3}{11} a^{11} + \frac{5}{11} a^{10} + \frac{4}{11} a^{9} + \frac{1}{11} a^{8} + \frac{3}{11} a^{7} - \frac{2}{11} a^{6} + \frac{2}{11} a^{5} - \frac{2}{11} a^{4} + \frac{5}{11} a^{3} + \frac{1}{11} a^{2} + \frac{3}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{17} + \frac{2}{11} a^{15} - \frac{4}{11} a^{14} - \frac{3}{11} a^{13} - \frac{1}{11} a^{12} - \frac{1}{11} a^{11} + \frac{5}{11} a^{10} + \frac{4}{11} a^{9} + \frac{1}{11} a^{8} + \frac{3}{11} a^{7} - \frac{5}{11} a^{6} + \frac{5}{11} a^{5} - \frac{2}{11} a^{4} + \frac{2}{11} a^{3} + \frac{1}{11} a^{2} + \frac{1}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{18} + \frac{3}{11} a^{15} - \frac{4}{11} a^{14} + \frac{5}{11} a^{13} - \frac{5}{11} a^{12} - \frac{1}{11} a^{11} + \frac{5}{11} a^{10} + \frac{4}{11} a^{9} + \frac{1}{11} a^{8} - \frac{2}{11} a^{6} + \frac{5}{11} a^{5} - \frac{5}{11} a^{4} + \frac{2}{11} a^{3} - \frac{1}{11} a^{2} + \frac{2}{11} a - \frac{3}{11}$, $\frac{1}{360694839079688200682542986536707} a^{19} - \frac{9624332784813477027833741450968}{360694839079688200682542986536707} a^{18} + \frac{1787630878560007115713104818800}{360694839079688200682542986536707} a^{17} - \frac{11866513358101109063019824980635}{360694839079688200682542986536707} a^{16} + \frac{128349758005409064555972702891}{300829724003076063955415334893} a^{15} - \frac{75948104277698181058784858087894}{360694839079688200682542986536707} a^{14} + \frac{71229948170855419917027542080112}{360694839079688200682542986536707} a^{13} + \frac{119429115069546082225276909980581}{360694839079688200682542986536707} a^{12} - \frac{34960281629654946269444917200429}{360694839079688200682542986536707} a^{11} - \frac{115295358232317044181836550480079}{360694839079688200682542986536707} a^{10} + \frac{122875109058157313726812201443639}{360694839079688200682542986536707} a^{9} + \frac{128448206378887804541001163307074}{360694839079688200682542986536707} a^{8} + \frac{48511091986004882299725398459860}{360694839079688200682542986536707} a^{7} + \frac{172878450802076285587178890169854}{360694839079688200682542986536707} a^{6} - \frac{144087015394430037811585940396384}{360694839079688200682542986536707} a^{5} + \frac{67395754248436889263856665590675}{360694839079688200682542986536707} a^{4} + \frac{58488520025645755158504667445176}{360694839079688200682542986536707} a^{3} - \frac{162040849915650439576334697427649}{360694839079688200682542986536707} a^{2} + \frac{144980802740397242911954330105972}{360694839079688200682542986536707} a + \frac{166181563299969838162286228609651}{360694839079688200682542986536707}$

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:  \( 2633227.55313 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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 $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
$2$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$