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Properties

Label 20.10.3098891881...0000.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{4}\cdot 5^{14}\cdot 19^{4}\cdot 29^{7}\cdot 109^{4}$
Root discriminant $53.03$
Ramified primes $2, 5, 19, 29, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T955

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7681, -19907, -47894, 194338, -37839, -493597, 513998, 200817, -495686, 70531, 194388, -71250, -36650, 20856, 2911, -2932, 25, 208, -17, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 - 17*x^18 + 208*x^17 + 25*x^16 - 2932*x^15 + 2911*x^14 + 20856*x^13 - 36650*x^12 - 71250*x^11 + 194388*x^10 + 70531*x^9 - 495686*x^8 + 200817*x^7 + 513998*x^6 - 493597*x^5 - 37839*x^4 + 194338*x^3 - 47894*x^2 - 19907*x + 7681)
 
gp: K = bnfinit(x^20 - 7*x^19 - 17*x^18 + 208*x^17 + 25*x^16 - 2932*x^15 + 2911*x^14 + 20856*x^13 - 36650*x^12 - 71250*x^11 + 194388*x^10 + 70531*x^9 - 495686*x^8 + 200817*x^7 + 513998*x^6 - 493597*x^5 - 37839*x^4 + 194338*x^3 - 47894*x^2 - 19907*x + 7681, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} - 17 x^{18} + 208 x^{17} + 25 x^{16} - 2932 x^{15} + 2911 x^{14} + 20856 x^{13} - 36650 x^{12} - 71250 x^{11} + 194388 x^{10} + 70531 x^{9} - 495686 x^{8} + 200817 x^{7} + 513998 x^{6} - 493597 x^{5} - 37839 x^{4} + 194338 x^{3} - 47894 x^{2} - 19907 x + 7681 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-30988918815000698609123925781250000=-\,2^{4}\cdot 5^{14}\cdot 19^{4}\cdot 29^{7}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.03$
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:  $2, 5, 19, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{10} a^{18} + \frac{1}{5} a^{16} + \frac{1}{5} a^{15} - \frac{3}{10} a^{14} - \frac{1}{2} a^{13} - \frac{1}{10} a^{12} + \frac{2}{5} a^{11} - \frac{1}{10} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{10} a^{5} - \frac{3}{10} a^{4} + \frac{3}{10} a^{3} - \frac{1}{2} a^{2} + \frac{1}{10}$, $\frac{1}{31164761432529256692465286309304269963922430} a^{19} + \frac{95544704007663860223631654897127343694042}{3116476143252925669246528630930426996392243} a^{18} + \frac{146136611200439495128058524241841143096004}{1731375635140514260692515906072459442440135} a^{17} + \frac{435884734282031526821642424214083278927456}{15582380716264628346232643154652134981961215} a^{16} + \frac{181978135111668337083754971747832621396873}{3462751270281028521385031812144918884880270} a^{15} - \frac{2720988304867536656806931002577182111366229}{6232952286505851338493057261860853992784486} a^{14} - \frac{12331248589829464404322405254436531528651171}{31164761432529256692465286309304269963922430} a^{13} - \frac{2574676936330453998076983586062420106988293}{15582380716264628346232643154652134981961215} a^{12} - \frac{488598972976212134621853133863753839960059}{3462751270281028521385031812144918884880270} a^{11} - \frac{1505027287266338367120862226909391197699747}{10388253810843085564155095436434756654640810} a^{10} + \frac{1179240057984591351445438104439492029208607}{5194126905421542782077547718217378327320405} a^{9} + \frac{6858610558890268738667052207866103080984063}{15582380716264628346232643154652134981961215} a^{8} + \frac{151827415548709274229139647893813082652054}{5194126905421542782077547718217378327320405} a^{7} + \frac{264281516684160453260585790661023201404203}{10388253810843085564155095436434756654640810} a^{6} - \frac{8917979375245687801926883683106702306647253}{31164761432529256692465286309304269963922430} a^{5} + \frac{854592709549809464003561982999317882661597}{3462751270281028521385031812144918884880270} a^{4} + \frac{76244199405294527787915038804074256113741}{2077650762168617112831019087286951330928162} a^{3} + \frac{185296287694308634298518588479094060530793}{3116476143252925669246528630930426996392243} a^{2} - \frac{4944829739848565700199061461714369866359363}{10388253810843085564155095436434756654640810} a + \frac{240512477103872307133150066187149855499506}{3116476143252925669246528630930426996392243}$

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:  $14$
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:  \( 20518554805.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.8172298511640625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R $20$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$109$109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$