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Properties

Label 20.0.40194758889...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 11^{10}$
Root discriminant $47.89$
Ramified primes $2, 3, 5, 11$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![247285, 87210, 13979, 364584, 94989, 5874, 254160, 18684, 57453, 42330, 22701, 5640, 11877, -432, 2790, -132, 381, -12, 29, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 29*x^18 - 12*x^17 + 381*x^16 - 132*x^15 + 2790*x^14 - 432*x^13 + 11877*x^12 + 5640*x^11 + 22701*x^10 + 42330*x^9 + 57453*x^8 + 18684*x^7 + 254160*x^6 + 5874*x^5 + 94989*x^4 + 364584*x^3 + 13979*x^2 + 87210*x + 247285)
 
gp: K = bnfinit(x^20 + 29*x^18 - 12*x^17 + 381*x^16 - 132*x^15 + 2790*x^14 - 432*x^13 + 11877*x^12 + 5640*x^11 + 22701*x^10 + 42330*x^9 + 57453*x^8 + 18684*x^7 + 254160*x^6 + 5874*x^5 + 94989*x^4 + 364584*x^3 + 13979*x^2 + 87210*x + 247285, 1)
 

Normalized defining polynomial

\( x^{20} + 29 x^{18} - 12 x^{17} + 381 x^{16} - 132 x^{15} + 2790 x^{14} - 432 x^{13} + 11877 x^{12} + 5640 x^{11} + 22701 x^{10} + 42330 x^{9} + 57453 x^{8} + 18684 x^{7} + 254160 x^{6} + 5874 x^{5} + 94989 x^{4} + 364584 x^{3} + 13979 x^{2} + 87210 x + 247285 \)

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:  \(4019475888928019955600000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.89$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{12} + \frac{1}{18} a^{8} + \frac{4}{9} a^{6} - \frac{1}{2} a^{4} + \frac{7}{18} a^{2} + \frac{1}{9}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{13} + \frac{1}{18} a^{9} + \frac{4}{9} a^{7} - \frac{1}{2} a^{5} + \frac{7}{18} a^{3} + \frac{1}{9} a$, $\frac{1}{36} a^{16} + \frac{1}{18} a^{12} - \frac{1}{18} a^{10} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} - \frac{2}{9} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{17}{36}$, $\frac{1}{36} a^{17} + \frac{1}{18} a^{13} - \frac{1}{18} a^{11} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{4}{9} a^{7} + \frac{1}{6} a^{6} + \frac{5}{18} a^{5} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{13}{36} a - \frac{1}{6}$, $\frac{1}{36} a^{18} - \frac{1}{12} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{5}{18}$, $\frac{1}{4489877806961476585050755722386232436866078116} a^{19} + \frac{643060665711957529561021686267884593921597}{118154679130565173290809361115427169391212582} a^{18} - \frac{21816396992772407573978857907163114879363419}{2244938903480738292525377861193116218433039058} a^{17} + \frac{13212487709538521195531782443731594712736369}{1122469451740369146262688930596558109216519529} a^{16} - \frac{3096532933249126179779890503889605147153179}{172687607960056791732721373937932016802541466} a^{15} + \frac{55795489406212784529188209495989791144709565}{2244938903480738292525377861193116218433039058} a^{14} - \frac{6287714081646999435007985809068513953269249}{86343803980028395866360686968966008401270733} a^{13} - \frac{9458432732448300415441388600216738574399210}{124718827971152127362520992288506456579613281} a^{12} + \frac{279577172823543745556316878356253097829131311}{4489877806961476585050755722386232436866078116} a^{11} + \frac{111116505732822168599095641391848071158599845}{2244938903480738292525377861193116218433039058} a^{10} + \frac{159351712556853355689266359597481806214731921}{2244938903480738292525377861193116218433039058} a^{9} - \frac{193480624902206380459431729206333255385280961}{2244938903480738292525377861193116218433039058} a^{8} + \frac{1061232915836296184700662707366888796718975925}{2244938903480738292525377861193116218433039058} a^{7} + \frac{59109368158658624996298580029784955831623597}{249437655942304254725041984577012913159226562} a^{6} + \frac{39720818203984594529431790798360409682844461}{2244938903480738292525377861193116218433039058} a^{5} + \frac{531226040647371502812892125648818337635448371}{2244938903480738292525377861193116218433039058} a^{4} + \frac{1949108570016160727635989510548147982473504531}{4489877806961476585050755722386232436866078116} a^{3} + \frac{734390208704650642166097166766033106204659}{3216244847393607868947532752425667934717821} a^{2} - \frac{516903263717008146883101673615254790691124371}{1122469451740369146262688930596558109216519529} a + \frac{27903310975671693491333501841963535854684388}{59077339565282586645404680557713584695606291}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 391894899.91517335 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-11}, \sqrt{-15})\), 5.1.162000.1, 10.0.393660000000.1, 10.0.4226622444000000.1, 10.2.63399336660000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
11Data not computed