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Properties

Label 20.0.10831531471...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{18}\cdot 37^{10}$
Root discriminant $44.85$
Ramified primes $3, 5, 37$
Class number $16$ (GRH)
Class group $[16]$ (GRH)
Galois group $D_5^2$ (as 20T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12321, -86247, 229548, -227550, -39506, 182237, 5435, -219873, 349654, -360176, 279549, -175591, 91806, -41161, 15921, -5252, 1498, -354, 71, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 71*x^18 - 354*x^17 + 1498*x^16 - 5252*x^15 + 15921*x^14 - 41161*x^13 + 91806*x^12 - 175591*x^11 + 279549*x^10 - 360176*x^9 + 349654*x^8 - 219873*x^7 + 5435*x^6 + 182237*x^5 - 39506*x^4 - 227550*x^3 + 229548*x^2 - 86247*x + 12321)
 
gp: K = bnfinit(x^20 - 10*x^19 + 71*x^18 - 354*x^17 + 1498*x^16 - 5252*x^15 + 15921*x^14 - 41161*x^13 + 91806*x^12 - 175591*x^11 + 279549*x^10 - 360176*x^9 + 349654*x^8 - 219873*x^7 + 5435*x^6 + 182237*x^5 - 39506*x^4 - 227550*x^3 + 229548*x^2 - 86247*x + 12321, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 71 x^{18} - 354 x^{17} + 1498 x^{16} - 5252 x^{15} + 15921 x^{14} - 41161 x^{13} + 91806 x^{12} - 175591 x^{11} + 279549 x^{10} - 360176 x^{9} + 349654 x^{8} - 219873 x^{7} + 5435 x^{6} + 182237 x^{5} - 39506 x^{4} - 227550 x^{3} + 229548 x^{2} - 86247 x + 12321 \)

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:  \(1083153147151571524051666259765625=3^{10}\cdot 5^{18}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.85$
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:  $3, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{1665} a^{16} - \frac{8}{1665} a^{15} - \frac{19}{555} a^{14} + \frac{539}{1665} a^{13} - \frac{142}{1665} a^{12} + \frac{47}{555} a^{11} + \frac{694}{1665} a^{10} + \frac{157}{1665} a^{9} - \frac{704}{1665} a^{8} - \frac{1}{5} a^{7} - \frac{688}{1665} a^{6} - \frac{358}{1665} a^{5} - \frac{52}{185} a^{4} + \frac{661}{1665} a^{3} - \frac{323}{1665} a^{2} - \frac{7}{15} a - \frac{1}{15}$, $\frac{1}{1665} a^{17} - \frac{121}{1665} a^{15} + \frac{83}{1665} a^{14} - \frac{55}{111} a^{13} + \frac{134}{333} a^{12} + \frac{157}{1665} a^{11} + \frac{238}{555} a^{10} + \frac{184}{555} a^{9} + \frac{139}{333} a^{8} - \frac{22}{1665} a^{7} + \frac{266}{555} a^{6} - \frac{2}{1665} a^{5} + \frac{247}{1665} a^{4} - \frac{2}{111} a^{3} - \frac{31}{1665} a^{2} + \frac{1}{5} a + \frac{7}{15}$, $\frac{1}{884627028143159874615} a^{18} - \frac{1}{98291892015906652735} a^{17} + \frac{245181081264902708}{884627028143159874615} a^{16} - \frac{392289730023844292}{176925405628631974923} a^{15} - \frac{16203407876969640218}{294875676047719958205} a^{14} + \frac{53513845256206974202}{126375289734737124945} a^{13} - \frac{7469663118914032097}{23908838598463780395} a^{12} + \frac{4662107707378341134}{17345628002807056365} a^{11} - \frac{1464686409576224654}{4401129493249551615} a^{10} + \frac{11622825890488816477}{52036884008421169095} a^{9} + \frac{323166017592265396784}{884627028143159874615} a^{8} + \frac{111615380163812879924}{294875676047719958205} a^{7} + \frac{36800725584815164481}{176925405628631974923} a^{6} - \frac{12226821275471358503}{126375289734737124945} a^{5} - \frac{37259950105854858832}{294875676047719958205} a^{4} + \frac{11550317223251502526}{25275057946947424989} a^{3} - \frac{17741729108960238908}{98291892015906652735} a^{2} - \frac{5603926726791427}{25625764842940815} a + \frac{863115134859235003}{2656537622051531155}$, $\frac{1}{4443481562363092050191145} a^{19} + \frac{278}{493720173595899116687905} a^{18} - \frac{715426054299417613048}{4443481562363092050191145} a^{17} + \frac{1666633957295478292}{296232104157539470012743} a^{16} + \frac{38846667442541457684442}{634783080337584578598735} a^{15} - \frac{206447289219692645199134}{4443481562363092050191145} a^{14} - \frac{246091284934679595521717}{493720173595899116687905} a^{13} - \frac{6911155167357722947334}{16518518819193650744205} a^{12} - \frac{600684908143690055389213}{1481160520787697350063715} a^{11} - \frac{420698388660628150905484}{1481160520787697350063715} a^{10} - \frac{77916525121350220728791}{1481160520787697350063715} a^{9} - \frac{2198911533736686413826131}{4443481562363092050191145} a^{8} - \frac{45891703013590024842569}{634783080337584578598735} a^{7} + \frac{33290522380862653165024}{87127089458099844121395} a^{6} + \frac{492727354089971604575294}{4443481562363092050191145} a^{5} - \frac{1097482234185229324731532}{4443481562363092050191145} a^{4} - \frac{1797468419499353857253594}{4443481562363092050191145} a^{3} + \frac{3820655982467342858002}{66320620333777493286435} a^{2} + \frac{1973896276121818282499}{40031365426694522974695} a - \frac{9578707222907711109847}{40031365426694522974695}$

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

Class group and class number

$C_{16}$, which has order $16$ (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:  \( 85002563.2373 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2$ (as 20T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 16 conjugacy class representatives for $D_5^2$
Character table for $D_5^2$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-111}) \), \(\Q(\sqrt{185}) \), \(\Q(\sqrt{-15}, \sqrt{-111})\), 10.0.6582258418359375.1 x5

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 sibling: data not computed
Degree 25 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.13.7$x^{10} + 5 x^{4} + 10$$10$$1$$13$$D_5$$[3/2]_{2}$
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$