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Properties

Label 20.0.90565194372...5625.6
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $88.68$
Ramified primes $3, 5, 7, 11$
Class number $126808$ (GRH)
Class group $[22, 5764]$ (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![507348601, -19105806, 266085953, -44927537, 65617877, -22665265, 14902647, -4741860, 2776705, -832912, 363197, -141048, 48113, -10092, 4746, -1764, 340, -42, 25, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 25*x^18 - 42*x^17 + 340*x^16 - 1764*x^15 + 4746*x^14 - 10092*x^13 + 48113*x^12 - 141048*x^11 + 363197*x^10 - 832912*x^9 + 2776705*x^8 - 4741860*x^7 + 14902647*x^6 - 22665265*x^5 + 65617877*x^4 - 44927537*x^3 + 266085953*x^2 - 19105806*x + 507348601)
 
gp: K = bnfinit(x^20 - 8*x^19 + 25*x^18 - 42*x^17 + 340*x^16 - 1764*x^15 + 4746*x^14 - 10092*x^13 + 48113*x^12 - 141048*x^11 + 363197*x^10 - 832912*x^9 + 2776705*x^8 - 4741860*x^7 + 14902647*x^6 - 22665265*x^5 + 65617877*x^4 - 44927537*x^3 + 266085953*x^2 - 19105806*x + 507348601, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 25 x^{18} - 42 x^{17} + 340 x^{16} - 1764 x^{15} + 4746 x^{14} - 10092 x^{13} + 48113 x^{12} - 141048 x^{11} + 363197 x^{10} - 832912 x^{9} + 2776705 x^{8} - 4741860 x^{7} + 14902647 x^{6} - 22665265 x^{5} + 65617877 x^{4} - 44927537 x^{3} + 266085953 x^{2} - 19105806 x + 507348601 \)

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:  \(905651943727436308698404364703916015625=3^{10}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.68$
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:  $3, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1155=3\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1155}(1,·)$, $\chi_{1155}(1154,·)$, $\chi_{1155}(391,·)$, $\chi_{1155}(841,·)$, $\chi_{1155}(1079,·)$, $\chi_{1155}(524,·)$, $\chi_{1155}(706,·)$, $\chi_{1155}(526,·)$, $\chi_{1155}(344,·)$, $\chi_{1155}(601,·)$, $\chi_{1155}(734,·)$, $\chi_{1155}(421,·)$, $\chi_{1155}(554,·)$, $\chi_{1155}(811,·)$, $\chi_{1155}(76,·)$, $\chi_{1155}(449,·)$, $\chi_{1155}(629,·)$, $\chi_{1155}(631,·)$, $\chi_{1155}(314,·)$, $\chi_{1155}(764,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{23} a^{10} + \frac{2}{23} a^{9} + \frac{6}{23} a^{8} - \frac{1}{23} a^{7} - \frac{11}{23} a^{6} - \frac{2}{23} a^{5} + \frac{8}{23} a^{4} + \frac{8}{23} a^{3} - \frac{9}{23} a^{2} + \frac{5}{23} a - \frac{7}{23}$, $\frac{1}{23} a^{11} + \frac{2}{23} a^{9} + \frac{10}{23} a^{8} - \frac{9}{23} a^{7} - \frac{3}{23} a^{6} - \frac{11}{23} a^{5} - \frac{8}{23} a^{4} - \frac{2}{23} a^{3} + \frac{6}{23} a - \frac{9}{23}$, $\frac{1}{23} a^{12} + \frac{6}{23} a^{9} + \frac{2}{23} a^{8} - \frac{1}{23} a^{7} + \frac{11}{23} a^{6} - \frac{4}{23} a^{5} + \frac{5}{23} a^{4} + \frac{7}{23} a^{3} + \frac{1}{23} a^{2} + \frac{4}{23} a - \frac{9}{23}$, $\frac{1}{23} a^{13} - \frac{10}{23} a^{9} + \frac{9}{23} a^{8} - \frac{6}{23} a^{7} - \frac{7}{23} a^{6} - \frac{6}{23} a^{5} + \frac{5}{23} a^{4} - \frac{1}{23} a^{3} - \frac{11}{23} a^{2} + \frac{7}{23} a - \frac{4}{23}$, $\frac{1}{23} a^{14} + \frac{6}{23} a^{9} + \frac{8}{23} a^{8} + \frac{6}{23} a^{7} - \frac{1}{23} a^{6} + \frac{8}{23} a^{5} + \frac{10}{23} a^{4} + \frac{9}{23} a^{2} - \frac{1}{23}$, $\frac{1}{23} a^{15} - \frac{4}{23} a^{9} - \frac{7}{23} a^{8} + \frac{5}{23} a^{7} + \frac{5}{23} a^{6} - \frac{1}{23} a^{5} - \frac{2}{23} a^{4} + \frac{7}{23} a^{3} + \frac{8}{23} a^{2} - \frac{8}{23} a - \frac{4}{23}$, $\frac{1}{23} a^{16} + \frac{1}{23} a^{9} + \frac{6}{23} a^{8} + \frac{1}{23} a^{7} + \frac{1}{23} a^{6} - \frac{10}{23} a^{5} - \frac{7}{23} a^{4} - \frac{6}{23} a^{3} + \frac{2}{23} a^{2} - \frac{7}{23} a - \frac{5}{23}$, $\frac{1}{529} a^{17} + \frac{1}{529} a^{16} + \frac{2}{529} a^{15} - \frac{8}{529} a^{14} - \frac{1}{529} a^{13} + \frac{5}{529} a^{12} + \frac{4}{529} a^{10} + \frac{20}{529} a^{9} + \frac{247}{529} a^{8} - \frac{38}{529} a^{7} - \frac{146}{529} a^{6} - \frac{126}{529} a^{5} - \frac{191}{529} a^{4} - \frac{137}{529} a^{3} + \frac{66}{529} a^{2} - \frac{7}{23} a - \frac{67}{529}$, $\frac{1}{529} a^{18} + \frac{1}{529} a^{16} - \frac{10}{529} a^{15} + \frac{7}{529} a^{14} + \frac{6}{529} a^{13} - \frac{5}{529} a^{12} + \frac{4}{529} a^{11} - \frac{7}{529} a^{10} + \frac{181}{529} a^{9} + \frac{106}{529} a^{8} - \frac{85}{529} a^{7} - \frac{256}{529} a^{6} - \frac{19}{529} a^{5} - \frac{130}{529} a^{4} + \frac{19}{529} a^{3} - \frac{20}{529} a^{2} - \frac{21}{529} a + \frac{228}{529}$, $\frac{1}{1263097401808916872832294340198707312129998554925096450201} a^{19} + \frac{717453190373278711121788498607557150718562483087256477}{1263097401808916872832294340198707312129998554925096450201} a^{18} - \frac{817046637504387119316891205214416864582999008166887006}{1263097401808916872832294340198707312129998554925096450201} a^{17} + \frac{2923051926363797922013336019997244653998400483147298954}{1263097401808916872832294340198707312129998554925096450201} a^{16} - \frac{24028691565509157884588151452530628449293369604769429844}{1263097401808916872832294340198707312129998554925096450201} a^{15} - \frac{10432469728401571555603450681616582908247880080283012384}{1263097401808916872832294340198707312129998554925096450201} a^{14} - \frac{14292740238192959858719764233219943755571275671489899757}{1263097401808916872832294340198707312129998554925096450201} a^{13} + \frac{4023100752748720509053623299379477693282120995653975069}{1263097401808916872832294340198707312129998554925096450201} a^{12} - \frac{15022428235879329391489042466060205717203357058763862820}{1263097401808916872832294340198707312129998554925096450201} a^{11} + \frac{2564758957623361481900701837132896816973439231289675963}{1263097401808916872832294340198707312129998554925096450201} a^{10} + \frac{192953469625275202338613260052994906271951255641046291596}{1263097401808916872832294340198707312129998554925096450201} a^{9} - \frac{625954408367154253117137050818050717649413371955158964664}{1263097401808916872832294340198707312129998554925096450201} a^{8} + \frac{360502170668774112583955601631891817163739832000733210413}{1263097401808916872832294340198707312129998554925096450201} a^{7} - \frac{184529770022064950703282688632726066582634368884219830945}{1263097401808916872832294340198707312129998554925096450201} a^{6} + \frac{543382195241048893048611280359002720234459991450263939554}{1263097401808916872832294340198707312129998554925096450201} a^{5} - \frac{299976310747767545512124768546931251954192252353151487853}{1263097401808916872832294340198707312129998554925096450201} a^{4} - \frac{331515074012688570847614050786765451679616674366218045871}{1263097401808916872832294340198707312129998554925096450201} a^{3} - \frac{570330037039261305466644333594925329929558315896536974302}{1263097401808916872832294340198707312129998554925096450201} a^{2} - \frac{478547245106613053258084286111948041748130660013283887576}{1263097401808916872832294340198707312129998554925096450201} a + \frac{569317583632454477764868389381537932341828374634285986997}{1263097401808916872832294340198707312129998554925096450201}$

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

Class group and class number

$C_{22}\times C_{5764}$, which has order $126808$ (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:  \( 1415140.16249 \) (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

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{-1155}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{77})\), \(\Q(\zeta_{11})^+\), 10.10.39630026842637.1, 10.0.30094051633627471875.1, 10.0.162778775259375.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
5Data not computed
7Data not computed
11Data not computed