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Properties

Label 20.0.56552251376...3664.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 29^{15}$
Root discriminant $21.76$
Ramified primes $2, 29$
Class number $1$
Class group Trivial
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![223, -1501, 5366, -13072, 23544, -32603, 35592, -31374, 23243, -15538, 10114, -6414, 3658, -1782, 780, -338, 147, -61, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 22*x^18 - 61*x^17 + 147*x^16 - 338*x^15 + 780*x^14 - 1782*x^13 + 3658*x^12 - 6414*x^11 + 10114*x^10 - 15538*x^9 + 23243*x^8 - 31374*x^7 + 35592*x^6 - 32603*x^5 + 23544*x^4 - 13072*x^3 + 5366*x^2 - 1501*x + 223)
 
gp: K = bnfinit(x^20 - 6*x^19 + 22*x^18 - 61*x^17 + 147*x^16 - 338*x^15 + 780*x^14 - 1782*x^13 + 3658*x^12 - 6414*x^11 + 10114*x^10 - 15538*x^9 + 23243*x^8 - 31374*x^7 + 35592*x^6 - 32603*x^5 + 23544*x^4 - 13072*x^3 + 5366*x^2 - 1501*x + 223, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 22 x^{18} - 61 x^{17} + 147 x^{16} - 338 x^{15} + 780 x^{14} - 1782 x^{13} + 3658 x^{12} - 6414 x^{11} + 10114 x^{10} - 15538 x^{9} + 23243 x^{8} - 31374 x^{7} + 35592 x^{6} - 32603 x^{5} + 23544 x^{4} - 13072 x^{3} + 5366 x^{2} - 1501 x + 223 \)

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:  \(565522513762594615522033664=2^{16}\cdot 29^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.76$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{8555} a^{16} - \frac{833}{8555} a^{15} - \frac{1462}{8555} a^{14} + \frac{123}{8555} a^{13} + \frac{502}{8555} a^{12} + \frac{3903}{8555} a^{11} - \frac{255}{1711} a^{10} + \frac{2817}{8555} a^{9} - \frac{4051}{8555} a^{8} + \frac{1154}{8555} a^{7} - \frac{3528}{8555} a^{6} - \frac{1261}{8555} a^{5} - \frac{2664}{8555} a^{4} - \frac{4122}{8555} a^{3} + \frac{347}{1711} a^{2} + \frac{616}{1711} a - \frac{3572}{8555}$, $\frac{1}{8555} a^{17} - \frac{137}{1711} a^{15} - \frac{1202}{8555} a^{14} + \frac{3723}{8555} a^{13} - \frac{794}{1711} a^{12} + \frac{147}{1711} a^{11} - \frac{3571}{8555} a^{10} + \frac{3573}{8555} a^{9} + \frac{837}{1711} a^{8} - \frac{14}{295} a^{7} + \frac{1124}{8555} a^{6} - \frac{146}{295} a^{5} + \frac{2777}{8555} a^{4} - \frac{1336}{8555} a^{3} - \frac{882}{8555} a^{2} - \frac{202}{1711} a - \frac{1758}{8555}$, $\frac{1}{8555} a^{18} - \frac{333}{8555} a^{15} + \frac{1477}{8555} a^{14} - \frac{132}{8555} a^{13} + \frac{694}{8555} a^{12} - \frac{887}{8555} a^{11} - \frac{614}{8555} a^{10} + \frac{3822}{8555} a^{9} - \frac{362}{1711} a^{8} - \frac{4001}{8555} a^{7} + \frac{1862}{8555} a^{6} - \frac{2086}{8555} a^{5} + \frac{2883}{8555} a^{4} - \frac{1302}{8555} a^{3} + \frac{1742}{8555} a^{2} + \frac{18}{1711} a + \frac{3332}{8555}$, $\frac{1}{6168239805715} a^{19} - \frac{325601124}{6168239805715} a^{18} + \frac{80540929}{6168239805715} a^{17} - \frac{30083941}{1233647961143} a^{16} + \frac{224974196762}{6168239805715} a^{15} + \frac{222380124778}{6168239805715} a^{14} - \frac{1410918334073}{6168239805715} a^{13} + \frac{327020702959}{1233647961143} a^{12} + \frac{574678866845}{1233647961143} a^{11} - \frac{613129739372}{6168239805715} a^{10} + \frac{1582384938731}{6168239805715} a^{9} + \frac{2063465574984}{6168239805715} a^{8} - \frac{2348208961356}{6168239805715} a^{7} - \frac{520861870309}{6168239805715} a^{6} + \frac{1234566086789}{6168239805715} a^{5} + \frac{2779066258624}{6168239805715} a^{4} + \frac{9216046308}{94895997011} a^{3} - \frac{130374445946}{1233647961143} a^{2} + \frac{1176298004631}{6168239805715} a - \frac{509469851431}{1233647961143}$

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:  \( 147782.929749 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.24389.1, 5.1.390224.1 x5, 10.2.4415968335104.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.390224.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{20}$

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
2Data not computed
$29$29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$