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Properties

Label 20.0.65712112720...3321.1
Degree $20$
Signature $[0, 10]$
Discriminant $19^{10}\cdot 41^{18}$
Root discriminant $123.28$
Ramified primes $19, 41$
Class number $1908775$ (GRH)
Class group $[11, 173525]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![785567709, -556910277, 748934741, -389384970, 312788154, -135622022, 80897637, -30276269, 14408932, -4738723, 1843943, -525211, 170951, -43261, 12405, -2698, 680, -146, 41, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 41*x^18 - 146*x^17 + 680*x^16 - 2698*x^15 + 12405*x^14 - 43261*x^13 + 170951*x^12 - 525211*x^11 + 1843943*x^10 - 4738723*x^9 + 14408932*x^8 - 30276269*x^7 + 80897637*x^6 - 135622022*x^5 + 312788154*x^4 - 389384970*x^3 + 748934741*x^2 - 556910277*x + 785567709)
 
gp: K = bnfinit(x^20 - 8*x^19 + 41*x^18 - 146*x^17 + 680*x^16 - 2698*x^15 + 12405*x^14 - 43261*x^13 + 170951*x^12 - 525211*x^11 + 1843943*x^10 - 4738723*x^9 + 14408932*x^8 - 30276269*x^7 + 80897637*x^6 - 135622022*x^5 + 312788154*x^4 - 389384970*x^3 + 748934741*x^2 - 556910277*x + 785567709, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 41 x^{18} - 146 x^{17} + 680 x^{16} - 2698 x^{15} + 12405 x^{14} - 43261 x^{13} + 170951 x^{12} - 525211 x^{11} + 1843943 x^{10} - 4738723 x^{9} + 14408932 x^{8} - 30276269 x^{7} + 80897637 x^{6} - 135622022 x^{5} + 312788154 x^{4} - 389384970 x^{3} + 748934741 x^{2} - 556910277 x + 785567709 \)

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:  \(657121127202216809347230953601268621783321=19^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.28$
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:  $19, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(779=19\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{779}(1,·)$, $\chi_{779}(324,·)$, $\chi_{779}(455,·)$, $\chi_{779}(778,·)$, $\chi_{779}(590,·)$, $\chi_{779}(400,·)$, $\chi_{779}(18,·)$, $\chi_{779}(666,·)$, $\chi_{779}(476,·)$, $\chi_{779}(474,·)$, $\chi_{779}(286,·)$, $\chi_{779}(37,·)$, $\chi_{779}(742,·)$, $\chi_{779}(113,·)$, $\chi_{779}(493,·)$, $\chi_{779}(303,·)$, $\chi_{779}(305,·)$, $\chi_{779}(761,·)$, $\chi_{779}(379,·)$, $\chi_{779}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{3} a^{8} - \frac{1}{9} a^{7} + \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{4}{9} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{8} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} - \frac{4}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{9} a^{6} - \frac{4}{9} a^{4} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{366777} a^{18} - \frac{6526}{366777} a^{17} - \frac{4364}{366777} a^{16} - \frac{7396}{366777} a^{15} + \frac{10475}{122259} a^{14} - \frac{2608}{40753} a^{13} - \frac{54707}{366777} a^{12} - \frac{16732}{122259} a^{11} - \frac{16352}{122259} a^{10} - \frac{60788}{366777} a^{9} + \frac{169574}{366777} a^{8} - \frac{9401}{366777} a^{7} + \frac{26534}{122259} a^{6} - \frac{48304}{122259} a^{5} - \frac{103090}{366777} a^{4} + \frac{53674}{122259} a^{3} - \frac{115165}{366777} a^{2} - \frac{56582}{122259} a - \frac{479}{40753}$, $\frac{1}{107422706768982348596493340302722519943048761878788836231307} a^{19} + \frac{9013114471953246469114789782827849954276579431273235}{11935856307664705399610371144746946660338751319865426247923} a^{18} + \frac{3494099737323930424614998276836247288591817051037245116492}{107422706768982348596493340302722519943048761878788836231307} a^{17} - \frac{3093133272290633171972437308297078652137620229415359636058}{107422706768982348596493340302722519943048761878788836231307} a^{16} + \frac{931569540717821207940790331356532582801336646230871002983}{35807568922994116198831113434240839981016253959596278743769} a^{15} + \frac{11494601355631052563434479507412636765502016916224056108544}{107422706768982348596493340302722519943048761878788836231307} a^{14} - \frac{7511089230281428906700348594985443776362770505833691431375}{107422706768982348596493340302722519943048761878788836231307} a^{13} + \frac{16962385102127087877445179388084303088188550607685677603358}{107422706768982348596493340302722519943048761878788836231307} a^{12} + \frac{3784850848503143701321473447678644077147031490386360158450}{107422706768982348596493340302722519943048761878788836231307} a^{11} - \frac{9201489168000973896919450881697771361690090050840527267995}{107422706768982348596493340302722519943048761878788836231307} a^{10} - \frac{8280578183797300869691158888572055477944749339727095049618}{107422706768982348596493340302722519943048761878788836231307} a^{9} - \frac{443187787933886195567147873225540898474600455813298185123}{1471543928342223953376621100037294793740393998339573099059} a^{8} + \frac{9858151275952231900064847198322203638501384386370569348957}{35807568922994116198831113434240839981016253959596278743769} a^{7} + \frac{12239047488492430832742844088796057215182923211172192721829}{107422706768982348596493340302722519943048761878788836231307} a^{6} - \frac{34835113357379394360750614419819311437695419834299599173271}{107422706768982348596493340302722519943048761878788836231307} a^{5} + \frac{12648480669827635847191642072076521974325071035036042229848}{107422706768982348596493340302722519943048761878788836231307} a^{4} + \frac{51134636599722945735243241566540259483331872277559368919613}{107422706768982348596493340302722519943048761878788836231307} a^{3} + \frac{35881558138910899644564893190952231910845485965826481165262}{107422706768982348596493340302722519943048761878788836231307} a^{2} - \frac{8839858877031737975184785728350442327860139302180903609332}{35807568922994116198831113434240839981016253959596278743769} a + \frac{4806514792312525160026520134715441297028120807336844481371}{11935856307664705399610371144746946660338751319865426247923}$

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

Class group and class number

$C_{11}\times C_{173525}$, which has order $1908775$ (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:  \( 5104264.636551031 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-779}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-19}, \sqrt{41})\), 5.5.2825761.1, 10.0.810630080370952438139.3, 10.10.327381934393961.1, 10.0.19771465374901278979.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
19Data not computed
41Data not computed