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Properties

Label 20.0.74851739808...5625.3
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 11^{18}\cdot 13^{10}$
Root discriminant $69.78$
Ramified primes $5, 11, 13$
Class number $69100$ (GRH)
Class group $[2, 34550]$ (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![471865021, -418237586, 559410178, -194745002, 208776667, -42951490, 58248347, -18471830, 17066125, -6455702, 3763442, -1288958, 545148, -154322, 50196, -11164, 2805, -452, 85, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 85*x^18 - 452*x^17 + 2805*x^16 - 11164*x^15 + 50196*x^14 - 154322*x^13 + 545148*x^12 - 1288958*x^11 + 3763442*x^10 - 6455702*x^9 + 17066125*x^8 - 18471830*x^7 + 58248347*x^6 - 42951490*x^5 + 208776667*x^4 - 194745002*x^3 + 559410178*x^2 - 418237586*x + 471865021)
 
gp: K = bnfinit(x^20 - 8*x^19 + 85*x^18 - 452*x^17 + 2805*x^16 - 11164*x^15 + 50196*x^14 - 154322*x^13 + 545148*x^12 - 1288958*x^11 + 3763442*x^10 - 6455702*x^9 + 17066125*x^8 - 18471830*x^7 + 58248347*x^6 - 42951490*x^5 + 208776667*x^4 - 194745002*x^3 + 559410178*x^2 - 418237586*x + 471865021, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 85 x^{18} - 452 x^{17} + 2805 x^{16} - 11164 x^{15} + 50196 x^{14} - 154322 x^{13} + 545148 x^{12} - 1288958 x^{11} + 3763442 x^{10} - 6455702 x^{9} + 17066125 x^{8} - 18471830 x^{7} + 58248347 x^{6} - 42951490 x^{5} + 208776667 x^{4} - 194745002 x^{3} + 559410178 x^{2} - 418237586 x + 471865021 \)

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:  \(7485173980890456751447848616884765625=5^{10}\cdot 11^{18}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.78$
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:  $5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(715=5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{715}(1,·)$, $\chi_{715}(194,·)$, $\chi_{715}(259,·)$, $\chi_{715}(196,·)$, $\chi_{715}(519,·)$, $\chi_{715}(456,·)$, $\chi_{715}(521,·)$, $\chi_{715}(586,·)$, $\chi_{715}(14,·)$, $\chi_{715}(144,·)$, $\chi_{715}(339,·)$, $\chi_{715}(599,·)$, $\chi_{715}(664,·)$, $\chi_{715}(129,·)$, $\chi_{715}(701,·)$, $\chi_{715}(51,·)$, $\chi_{715}(116,·)$, $\chi_{715}(376,·)$, $\chi_{715}(571,·)$, $\chi_{715}(714,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{5366} a^{12} - \frac{143}{5366} a^{11} - \frac{273}{2683} a^{10} - \frac{1679}{5366} a^{9} + \frac{1157}{5366} a^{8} + \frac{841}{2683} a^{7} + \frac{127}{5366} a^{6} - \frac{2229}{5366} a^{5} + \frac{1277}{2683} a^{4} - \frac{2637}{5366} a^{3} - \frac{1213}{2683} a^{2} + \frac{1242}{2683} a + \frac{795}{5366}$, $\frac{1}{5366} a^{13} + \frac{469}{5366} a^{11} + \frac{733}{5366} a^{10} + \frac{1265}{2683} a^{9} + \frac{787}{5366} a^{8} - \frac{817}{5366} a^{7} - \frac{83}{2683} a^{6} + \frac{401}{5366} a^{5} - \frac{2303}{5366} a^{4} + \frac{1469}{5366} a^{3} - \frac{505}{2683} a^{2} + \frac{1851}{5366} a + \frac{999}{5366}$, $\frac{1}{5366} a^{14} + \frac{725}{5366} a^{11} + \frac{518}{2683} a^{10} - \frac{282}{2683} a^{9} + \frac{1199}{5366} a^{8} - \frac{111}{2683} a^{7} - \frac{68}{2683} a^{6} - \frac{589}{5366} a^{5} + \frac{261}{5366} a^{4} + \frac{1563}{5366} a^{3} - \frac{315}{2683} a^{2} - \frac{1129}{2683} a + \frac{41}{2683}$, $\frac{1}{5366} a^{15} + \frac{37}{2683} a^{11} + \frac{885}{5366} a^{10} - \frac{2291}{5366} a^{9} - \frac{1951}{5366} a^{8} - \frac{752}{2683} a^{7} - \frac{721}{2683} a^{6} - \frac{1563}{5366} a^{5} - \frac{750}{2683} a^{4} - \frac{892}{2683} a^{3} + \frac{955}{2683} a^{2} - \frac{525}{5366} a - \frac{2213}{5366}$, $\frac{1}{5366} a^{16} + \frac{735}{5366} a^{11} + \frac{551}{5366} a^{10} - \frac{1123}{5366} a^{9} - \frac{633}{2683} a^{8} - \frac{1246}{2683} a^{7} - \frac{229}{5366} a^{6} + \frac{1233}{2683} a^{5} + \frac{1198}{2683} a^{4} - \frac{747}{2683} a^{3} + \frac{1921}{5366} a^{2} + \frac{1781}{5366} a + \frac{98}{2683}$, $\frac{1}{5366} a^{17} + \frac{1019}{5366} a^{11} + \frac{210}{2683} a^{10} + \frac{651}{2683} a^{9} + \frac{307}{5366} a^{8} - \frac{2319}{5366} a^{7} + \frac{343}{5366} a^{6} + \frac{699}{2683} a^{5} + \frac{2099}{5366} a^{4} + \frac{307}{5366} a^{3} - \frac{1987}{5366} a^{2} + \frac{1579}{5366} a + \frac{569}{5366}$, $\frac{1}{5366} a^{18} + \frac{1255}{5366} a^{11} - \frac{194}{2683} a^{10} - \frac{273}{2683} a^{9} - \frac{391}{2683} a^{8} - \frac{1861}{5366} a^{7} + \frac{769}{5366} a^{6} - \frac{867}{2683} a^{5} + \frac{291}{5366} a^{4} + \frac{1058}{2683} a^{3} - \frac{53}{5366} a^{2} + \frac{2125}{5366} a + \frac{161}{5366}$, $\frac{1}{285747246766660294056629897282712195300939590987271782} a^{19} - \frac{8530599018117329257189193069303242197054985031733}{285747246766660294056629897282712195300939590987271782} a^{18} + \frac{5330764430069479563803427653882592619151164062384}{142873623383330147028314948641356097650469795493635891} a^{17} - \frac{2153478503365346244784003525093015772461512435003}{285747246766660294056629897282712195300939590987271782} a^{16} + \frac{3669698394328690284487251080745911718956834591240}{142873623383330147028314948641356097650469795493635891} a^{15} + \frac{25750804349056129653997311156968325687333433193333}{285747246766660294056629897282712195300939590987271782} a^{14} - \frac{2646338553901680957200103314847980407856731283229}{285747246766660294056629897282712195300939590987271782} a^{13} + \frac{25686027423307911472230077849436070419380523046183}{285747246766660294056629897282712195300939590987271782} a^{12} - \frac{1501563004699553714019294750204689435488516212878813}{142873623383330147028314948641356097650469795493635891} a^{11} + \frac{3302996988665237294526752986910874005794297675335387}{142873623383330147028314948641356097650469795493635891} a^{10} + \frac{104802990748809827848069407421928327620659192310328433}{285747246766660294056629897282712195300939590987271782} a^{9} - \frac{129752691211726517163791869326246269965426155155443525}{285747246766660294056629897282712195300939590987271782} a^{8} + \frac{51155073596460612924500459529945542892121404976109023}{285747246766660294056629897282712195300939590987271782} a^{7} - \frac{45841623023595303310333929535905579841875967152631863}{142873623383330147028314948641356097650469795493635891} a^{6} + \frac{125024569949669137919942750786057617564475707993386357}{285747246766660294056629897282712195300939590987271782} a^{5} + \frac{115420919459203527801884536463145885353595543005229661}{285747246766660294056629897282712195300939590987271782} a^{4} - \frac{32738609063276751907940520264250754136284290282660617}{285747246766660294056629897282712195300939590987271782} a^{3} - \frac{64872979412281199206257168761666272926676530970635289}{285747246766660294056629897282712195300939590987271782} a^{2} + \frac{31827121935716235775629588937865806164374921001939370}{142873623383330147028314948641356097650469795493635891} a - \frac{56786332930153342941421329311554212212942949941299376}{142873623383330147028314948641356097650469795493635891}$

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

Class group and class number

$C_{2}\times C_{34550}$, which has order $69100$ (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:  \( 140644.599182 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-143}) \), \(\Q(\sqrt{-715}) \), \(\Q(\sqrt{5}, \sqrt{-143})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.875489472034463.1, 10.0.2735904600107696875.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
13Data not computed