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Properties

Label 20.0.13169261490...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{10}\cdot 3469^{5}$
Root discriminant $45.29$
Ramified primes $2, 5, 3469$
Class number $198$ (GRH)
Class group $[198]$ (GRH)
Galois group $C_2\times D_5\wr C_2$ (as 20T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![94025, 37050, 337065, 16550, 268819, -30316, 21318, -19136, -18513, 7432, 5177, -8670, 4905, -1546, 894, -808, 521, -216, 59, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 59*x^18 - 216*x^17 + 521*x^16 - 808*x^15 + 894*x^14 - 1546*x^13 + 4905*x^12 - 8670*x^11 + 5177*x^10 + 7432*x^9 - 18513*x^8 - 19136*x^7 + 21318*x^6 - 30316*x^5 + 268819*x^4 + 16550*x^3 + 337065*x^2 + 37050*x + 94025)
 
gp: K = bnfinit(x^20 - 10*x^19 + 59*x^18 - 216*x^17 + 521*x^16 - 808*x^15 + 894*x^14 - 1546*x^13 + 4905*x^12 - 8670*x^11 + 5177*x^10 + 7432*x^9 - 18513*x^8 - 19136*x^7 + 21318*x^6 - 30316*x^5 + 268819*x^4 + 16550*x^3 + 337065*x^2 + 37050*x + 94025, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 59 x^{18} - 216 x^{17} + 521 x^{16} - 808 x^{15} + 894 x^{14} - 1546 x^{13} + 4905 x^{12} - 8670 x^{11} + 5177 x^{10} + 7432 x^{9} - 18513 x^{8} - 19136 x^{7} + 21318 x^{6} - 30316 x^{5} + 268819 x^{4} + 16550 x^{3} + 337065 x^{2} + 37050 x + 94025 \)

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:  \(1316926149061478577602560000000000=2^{28}\cdot 5^{10}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.29$
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:  $2, 5, 3469$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{13} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{13} - \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{115} a^{18} - \frac{7}{115} a^{17} - \frac{1}{23} a^{16} + \frac{1}{115} a^{15} + \frac{3}{115} a^{14} - \frac{14}{115} a^{13} - \frac{13}{115} a^{12} + \frac{52}{115} a^{11} - \frac{38}{115} a^{10} + \frac{4}{23} a^{9} - \frac{52}{115} a^{8} - \frac{47}{115} a^{7} + \frac{27}{115} a^{6} + \frac{9}{115} a^{5} + \frac{36}{115} a^{4} + \frac{41}{115} a^{3} + \frac{42}{115} a^{2} + \frac{11}{23} a + \frac{3}{23}$, $\frac{1}{48561915479801838025648477002958945517910052327987495} a^{19} - \frac{118539284053367554012199997455362306130003130837192}{48561915479801838025648477002958945517910052327987495} a^{18} + \frac{1071468215002744377913762515680531022203204509192488}{48561915479801838025648477002958945517910052327987495} a^{17} - \frac{2348271760241632702212209855840088005635052448167988}{48561915479801838025648477002958945517910052327987495} a^{16} - \frac{2767622959158619289098478824908117636901090077900041}{48561915479801838025648477002958945517910052327987495} a^{15} - \frac{767571423244717012700257088773968501452489574138147}{48561915479801838025648477002958945517910052327987495} a^{14} + \frac{10723803224006928825213457512415588434212996649877442}{48561915479801838025648477002958945517910052327987495} a^{13} - \frac{1189787599490525744548908304873010899233173325174901}{9712383095960367605129695400591789103582010465597499} a^{12} - \frac{2250304280898363481287953518801160383043227896877512}{9712383095960367605129695400591789103582010465597499} a^{11} - \frac{18611585729944197933772372634199359384073364032136026}{48561915479801838025648477002958945517910052327987495} a^{10} + \frac{21191214697286950447821706413531912214206253584949427}{48561915479801838025648477002958945517910052327987495} a^{9} + \frac{14834292728064640223909053117018762584508768986771124}{48561915479801838025648477002958945517910052327987495} a^{8} + \frac{3689281309792266094205094410200803221956475703638367}{48561915479801838025648477002958945517910052327987495} a^{7} - \frac{22778114517903922345896882247352769261509237206571923}{48561915479801838025648477002958945517910052327987495} a^{6} - \frac{5012482566949649369928774488472854801823454596686253}{48561915479801838025648477002958945517910052327987495} a^{5} - \frac{152352637720807719599322886523581044735395790168355}{422277525911320330657812843503990830590522194156413} a^{4} - \frac{11131462723918239578683238128384651025588473605335617}{48561915479801838025648477002958945517910052327987495} a^{3} - \frac{1272152568579656591872548115152860689754081892315444}{48561915479801838025648477002958945517910052327987495} a^{2} + \frac{199949112068547456271178430736688531808388201000144}{9712383095960367605129695400591789103582010465597499} a + \frac{4650571806605325380713371434111322491619084994109814}{9712383095960367605129695400591789103582010465597499}$

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

Class group and class number

$C_{198}$, which has order $198$ (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:  \( 828338.933858 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_5\wr C_2$ (as 20T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 400
The 28 conjugacy class representatives for $C_2\times D_5\wr C_2$
Character table for $C_2\times D_5\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1387600.1, 10.10.9627168800000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3469Data not computed