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Properties

Label 20.0.37355077379...4048.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 17^{13}\cdot 37^{8}$
Root discriminant $75.62$
Ramified primes $2, 17, 37$
Class number $80$ (GRH)
Class group $[4, 20]$ (GRH)
Galois group $C_2^2:F_5$ (as 20T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![971137, -294146, 332021, -839814, 806930, -349100, 8409, 135816, -71994, -34470, 60110, -18634, -8760, 5050, 1416, -1220, 26, 108, -9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 9*x^18 + 108*x^17 + 26*x^16 - 1220*x^15 + 1416*x^14 + 5050*x^13 - 8760*x^12 - 18634*x^11 + 60110*x^10 - 34470*x^9 - 71994*x^8 + 135816*x^7 + 8409*x^6 - 349100*x^5 + 806930*x^4 - 839814*x^3 + 332021*x^2 - 294146*x + 971137)
 
gp: K = bnfinit(x^20 - 6*x^19 - 9*x^18 + 108*x^17 + 26*x^16 - 1220*x^15 + 1416*x^14 + 5050*x^13 - 8760*x^12 - 18634*x^11 + 60110*x^10 - 34470*x^9 - 71994*x^8 + 135816*x^7 + 8409*x^6 - 349100*x^5 + 806930*x^4 - 839814*x^3 + 332021*x^2 - 294146*x + 971137, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 9 x^{18} + 108 x^{17} + 26 x^{16} - 1220 x^{15} + 1416 x^{14} + 5050 x^{13} - 8760 x^{12} - 18634 x^{11} + 60110 x^{10} - 34470 x^{9} - 71994 x^{8} + 135816 x^{7} + 8409 x^{6} - 349100 x^{5} + 806930 x^{4} - 839814 x^{3} + 332021 x^{2} - 294146 x + 971137 \)

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:  \(37355077379825360472500371060108034048=2^{30}\cdot 17^{13}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.62$
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:  $2, 17, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{34} a^{16} - \frac{5}{34} a^{15} - \frac{4}{17} a^{14} + \frac{1}{34} a^{13} - \frac{1}{2} a^{12} - \frac{11}{34} a^{11} + \frac{1}{34} a^{10} + \frac{3}{34} a^{9} - \frac{1}{17} a^{8} + \frac{7}{34} a^{7} + \frac{3}{17} a^{6} + \frac{1}{17} a^{5} - \frac{15}{34} a^{3} + \frac{5}{34} a^{2} - \frac{7}{34} a + \frac{6}{17}$, $\frac{1}{34} a^{17} + \frac{1}{34} a^{15} - \frac{5}{34} a^{14} - \frac{6}{17} a^{13} + \frac{3}{17} a^{12} + \frac{7}{17} a^{11} + \frac{4}{17} a^{10} + \frac{13}{34} a^{9} - \frac{3}{34} a^{8} + \frac{7}{34} a^{7} - \frac{1}{17} a^{6} + \frac{5}{17} a^{5} - \frac{15}{34} a^{4} - \frac{1}{17} a^{3} - \frac{8}{17} a^{2} + \frac{11}{34} a - \frac{4}{17}$, $\frac{1}{306} a^{18} + \frac{1}{153} a^{17} - \frac{2}{153} a^{16} - \frac{7}{34} a^{15} - \frac{101}{306} a^{14} - \frac{19}{102} a^{13} - \frac{4}{153} a^{12} + \frac{1}{51} a^{11} + \frac{7}{306} a^{10} - \frac{77}{306} a^{9} + \frac{31}{153} a^{8} + \frac{11}{306} a^{7} - \frac{25}{102} a^{6} + \frac{131}{306} a^{5} - \frac{16}{153} a^{4} - \frac{13}{306} a^{3} - \frac{131}{306} a^{2} - \frac{1}{153} a + \frac{77}{306}$, $\frac{1}{3397717651018728483249608803040304263531928362767573602} a^{19} - \frac{2708273335096521933714529931582775425357922855670415}{1698858825509364241624804401520152131765964181383786801} a^{18} - \frac{22259624720237172734642586390913028589790349320673477}{1698858825509364241624804401520152131765964181383786801} a^{17} + \frac{620364505325989449093302304421649760542439149533039}{188762091723262693513867155724461347973996020153754089} a^{16} + \frac{518069777539147711874960431855038122876510642659914067}{3397717651018728483249608803040304263531928362767573602} a^{15} + \frac{220533625618837813248824153393585824781816716478466154}{566286275169788080541601467173384043921988060461262267} a^{14} - \frac{1375660676654852856253073669849790893146848647397318635}{3397717651018728483249608803040304263531928362767573602} a^{13} + \frac{14603332055769802096299846696174475571186786240910745}{566286275169788080541601467173384043921988060461262267} a^{12} - \frac{1525840015247184492631359722532111054801676437832330205}{3397717651018728483249608803040304263531928362767573602} a^{11} - \frac{1323409189913133830164469175288211959540752514937508663}{3397717651018728483249608803040304263531928362767573602} a^{10} + \frac{222107721507791867847049706767468739456626258535676931}{1698858825509364241624804401520152131765964181383786801} a^{9} + \frac{849310491899524259522855993326231332894329644402055671}{1698858825509364241624804401520152131765964181383786801} a^{8} + \frac{104139311814146786205442271555061604978826600676880876}{566286275169788080541601467173384043921988060461262267} a^{7} + \frac{452496134438970206800218775692948887682873142187136439}{1698858825509364241624804401520152131765964181383786801} a^{6} - \frac{56047480134818471204053879409648975030838890129644546}{1698858825509364241624804401520152131765964181383786801} a^{5} - \frac{833888395862672018225203106188680116676284067290249947}{3397717651018728483249608803040304263531928362767573602} a^{4} + \frac{379985400172572109476140136038301585117550714959580376}{1698858825509364241624804401520152131765964181383786801} a^{3} + \frac{24802647893134381504139881466199004748520916086707336}{1698858825509364241624804401520152131765964181383786801} a^{2} + \frac{147389761464169664336691282031011536802169870649427599}{3397717651018728483249608803040304263531928362767573602} a - \frac{110831961911152209630313258699140732950499570309838087}{377524183446525387027734311448922695947992040307508178}$

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

Class group and class number

$C_{4}\times C_{20}$, which has order $80$ (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:  \( 431927236.362 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.1088.2, 5.5.6725897.1, 10.10.1482348640816627712.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$