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Properties

Label 20.4.13167225195...4624.4
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{16}\cdot 23^{4}$
Root discriminant $18.03$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -18, 174, -523, 789, -608, 184, 18, 104, -105, 10, -55, 12, 54, -23, -22, 8, 7, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 4*x^18 + 7*x^17 + 8*x^16 - 22*x^15 - 23*x^14 + 54*x^13 + 12*x^12 - 55*x^11 + 10*x^10 - 105*x^9 + 104*x^8 + 18*x^7 + 184*x^6 - 608*x^5 + 789*x^4 - 523*x^3 + 174*x^2 - 18*x - 1)
 
gp: K = bnfinit(x^20 - x^19 - 4*x^18 + 7*x^17 + 8*x^16 - 22*x^15 - 23*x^14 + 54*x^13 + 12*x^12 - 55*x^11 + 10*x^10 - 105*x^9 + 104*x^8 + 18*x^7 + 184*x^6 - 608*x^5 + 789*x^4 - 523*x^3 + 174*x^2 - 18*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 4 x^{18} + 7 x^{17} + 8 x^{16} - 22 x^{15} - 23 x^{14} + 54 x^{13} + 12 x^{12} - 55 x^{11} + 10 x^{10} - 105 x^{9} + 104 x^{8} + 18 x^{7} + 184 x^{6} - 608 x^{5} + 789 x^{4} - 523 x^{3} + 174 x^{2} - 18 x - 1 \)

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:  \(13167225195265942636954624=2^{10}\cdot 11^{16}\cdot 23^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.03$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{3}{11} a^{14} - \frac{5}{11} a^{13} - \frac{2}{11} a^{12} - \frac{3}{11} a^{11} + \frac{2}{11} a^{10} + \frac{5}{11} a^{9} - \frac{1}{11} a^{8} + \frac{2}{11} a^{7} - \frac{3}{11} a^{6} + \frac{4}{11} a^{4} - \frac{5}{11} a^{3} - \frac{3}{11} a^{2} + \frac{5}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{3}{11} a^{14} + \frac{2}{11} a^{13} + \frac{3}{11} a^{12} - \frac{1}{11} a^{10} - \frac{5}{11} a^{9} + \frac{5}{11} a^{8} + \frac{2}{11} a^{7} - \frac{2}{11} a^{6} + \frac{4}{11} a^{5} + \frac{5}{11} a^{4} + \frac{1}{11} a^{3} + \frac{3}{11} a^{2} - \frac{3}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{17} - \frac{1}{11} a^{13} + \frac{5}{11} a^{12} + \frac{1}{11} a^{11} + \frac{1}{11} a^{10} - \frac{2}{11} a^{9} - \frac{1}{11} a^{8} + \frac{4}{11} a^{7} - \frac{5}{11} a^{6} + \frac{5}{11} a^{5} + \frac{2}{11} a^{4} - \frac{1}{11} a^{3} - \frac{1}{11} a^{2} + \frac{1}{11} a + \frac{3}{11}$, $\frac{1}{473} a^{18} - \frac{12}{473} a^{17} - \frac{3}{473} a^{16} + \frac{21}{473} a^{15} + \frac{170}{473} a^{14} - \frac{171}{473} a^{13} + \frac{9}{43} a^{12} + \frac{124}{473} a^{11} - \frac{2}{473} a^{10} - \frac{6}{43} a^{9} - \frac{163}{473} a^{8} - \frac{72}{473} a^{7} - \frac{25}{473} a^{6} + \frac{7}{473} a^{5} - \frac{17}{43} a^{4} + \frac{145}{473} a^{3} + \frac{84}{473} a^{2} - \frac{60}{473} a + \frac{236}{473}$, $\frac{1}{2234985931442180653} a^{19} - \frac{1383960253294078}{2234985931442180653} a^{18} + \frac{89489078887384375}{2234985931442180653} a^{17} + \frac{5222957276254252}{203180539222016423} a^{16} - \frac{31131077379953349}{2234985931442180653} a^{15} + \frac{339380196693620672}{2234985931442180653} a^{14} - \frac{496995957780189212}{2234985931442180653} a^{13} + \frac{922951404314967751}{2234985931442180653} a^{12} + \frac{480014248185742080}{2234985931442180653} a^{11} + \frac{185268367756392476}{2234985931442180653} a^{10} + \frac{558045316852257655}{2234985931442180653} a^{9} + \frac{321658149090036646}{2234985931442180653} a^{8} + \frac{761309260859444064}{2234985931442180653} a^{7} + \frac{670957422820655050}{2234985931442180653} a^{6} - \frac{398949788247417465}{2234985931442180653} a^{5} - \frac{25900718199544570}{51976417010283271} a^{4} + \frac{276502061826057084}{2234985931442180653} a^{3} - \frac{68245320937257583}{203180539222016423} a^{2} + \frac{17123874180424529}{51976417010283271} a + \frac{109343042227801599}{2234985931442180653}$

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:  \( 32385.6580745 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.4930254263.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{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}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$