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Properties

Label 20.4.75134130819...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{22}\cdot 37^{10}$
Root discriminant $62.20$
Ramified primes $2, 5, 37$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9909563804, 204431220, -8583237940, -44093080, 3420606480, -1767742, -833762140, 1405990, 138293980, -179820, -16314455, 10530, 1383545, -210, -83140, -4, 3390, 0, -85, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 85*x^18 + 3390*x^16 - 4*x^15 - 83140*x^14 - 210*x^13 + 1383545*x^12 + 10530*x^11 - 16314455*x^10 - 179820*x^9 + 138293980*x^8 + 1405990*x^7 - 833762140*x^6 - 1767742*x^5 + 3420606480*x^4 - 44093080*x^3 - 8583237940*x^2 + 204431220*x + 9909563804)
 
gp: K = bnfinit(x^20 - 85*x^18 + 3390*x^16 - 4*x^15 - 83140*x^14 - 210*x^13 + 1383545*x^12 + 10530*x^11 - 16314455*x^10 - 179820*x^9 + 138293980*x^8 + 1405990*x^7 - 833762140*x^6 - 1767742*x^5 + 3420606480*x^4 - 44093080*x^3 - 8583237940*x^2 + 204431220*x + 9909563804, 1)
 

Normalized defining polynomial

\( x^{20} - 85 x^{18} + 3390 x^{16} - 4 x^{15} - 83140 x^{14} - 210 x^{13} + 1383545 x^{12} + 10530 x^{11} - 16314455 x^{10} - 179820 x^{9} + 138293980 x^{8} + 1405990 x^{7} - 833762140 x^{6} - 1767742 x^{5} + 3420606480 x^{4} - 44093080 x^{3} - 8583237940 x^{2} + 204431220 x + 9909563804 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(751341308190288906250000000000000000=2^{16}\cdot 5^{22}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.20$
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:  $2, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{1}{14} a^{16} - \frac{3}{14} a^{15} - \frac{3}{14} a^{14} - \frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{14} a^{11} - \frac{1}{7} a^{10} + \frac{1}{14} a^{9} + \frac{1}{14} a^{8} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{3555194963118173635475830416440258902713453116503048307847442092772502} a^{19} - \frac{62844882517117173760561411441756274362645587203757462982422251307985}{3555194963118173635475830416440258902713453116503048307847442092772502} a^{18} + \frac{5636952143199034846600835767164130631065451040369180707526267410723}{26730789196377245379517521928122247388822955763180814344717609720094} a^{17} - \frac{5459604884712361712960675042308817045562491652661056159443995303246}{93557762187320358828311326748427865860880345171132850206511634020329} a^{16} + \frac{237561560521079318670638067014331673903989312931604598803927471533460}{1777597481559086817737915208220129451356726558251524153923721046386251} a^{15} + \frac{390401788888432181093457302789565369879777841607675644211684256237716}{1777597481559086817737915208220129451356726558251524153923721046386251} a^{14} - \frac{215359016795250546286025983634036988213448977103855374831322717384676}{1777597481559086817737915208220129451356726558251524153923721046386251} a^{13} + \frac{189958385483834018944591510764764770723813378183289023314311651587022}{1777597481559086817737915208220129451356726558251524153923721046386251} a^{12} - \frac{2946787591111275181545926642325307291334385001099571687021471460788}{13365394598188622689758760964061123694411477881590407172358804860047} a^{11} + \frac{102120469990507659580952812504708970983964287900889546448736447412329}{3555194963118173635475830416440258902713453116503048307847442092772502} a^{10} - \frac{745915627261555519379022809605898927593666270800969736050475767100278}{1777597481559086817737915208220129451356726558251524153923721046386251} a^{9} - \frac{648579953925530651659514362864485206952082907212195301586891765194590}{1777597481559086817737915208220129451356726558251524153923721046386251} a^{8} - \frac{419295850307387997592661761729549627250072695490591721850736176070279}{3555194963118173635475830416440258902713453116503048307847442092772502} a^{7} - \frac{411510458838320295910458152337287292340576199813707492603215874717144}{1777597481559086817737915208220129451356726558251524153923721046386251} a^{6} + \frac{1048855666324642203948755792811752133867536528809429445856928215605475}{3555194963118173635475830416440258902713453116503048307847442092772502} a^{5} - \frac{179963789477443889367395482641222486871086445535332508497475886703}{858328093461654668149645199526861154686975643771861011068914073581} a^{4} - \frac{7019541559090396900154557361466611792313540669316528188995011684921}{93557762187320358828311326748427865860880345171132850206511634020329} a^{3} + \frac{783940926583817673919301856106776378423631585279599703993184349295879}{1777597481559086817737915208220129451356726558251524153923721046386251} a^{2} - \frac{7430069184987517385613365724157206527156388613963318334373993933567}{93557762187320358828311326748427865860880345171132850206511634020329} a + \frac{41958924145982978936909759381798801510842706187344999016357502168635}{253942497365583831105416458317161350193818079750217736274817292340893}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{185}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{37})\), 5.1.50000.1, 10.2.173359892500000000.1, 10.2.866799462500000000.1, 10.2.12500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$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}$
5Data not computed
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$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}$