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Properties

Label 20.0.90565194372...625.10
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 $760848$ (GRH)
Class group $[2, 22, 17292]$ (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![20879692219, -11050362012, 17116956876, -3683482980, 5597098419, -1067010636, 1418808603, -381754838, 288415643, -80865818, 39686780, -9780738, 3525510, -712242, 198784, -31184, 6831, -760, 129, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 129*x^18 - 760*x^17 + 6831*x^16 - 31184*x^15 + 198784*x^14 - 712242*x^13 + 3525510*x^12 - 9780738*x^11 + 39686780*x^10 - 80865818*x^9 + 288415643*x^8 - 381754838*x^7 + 1418808603*x^6 - 1067010636*x^5 + 5597098419*x^4 - 3683482980*x^3 + 17116956876*x^2 - 11050362012*x + 20879692219)
 
gp: K = bnfinit(x^20 - 8*x^19 + 129*x^18 - 760*x^17 + 6831*x^16 - 31184*x^15 + 198784*x^14 - 712242*x^13 + 3525510*x^12 - 9780738*x^11 + 39686780*x^10 - 80865818*x^9 + 288415643*x^8 - 381754838*x^7 + 1418808603*x^6 - 1067010636*x^5 + 5597098419*x^4 - 3683482980*x^3 + 17116956876*x^2 - 11050362012*x + 20879692219, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 129 x^{18} - 760 x^{17} + 6831 x^{16} - 31184 x^{15} + 198784 x^{14} - 712242 x^{13} + 3525510 x^{12} - 9780738 x^{11} + 39686780 x^{10} - 80865818 x^{9} + 288415643 x^{8} - 381754838 x^{7} + 1418808603 x^{6} - 1067010636 x^{5} + 5597098419 x^{4} - 3683482980 x^{3} + 17116956876 x^{2} - 11050362012 x + 20879692219 \)

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$$\chi_{1155}(64,·)$, $\chi_{1155}(1,·)$, $\chi_{1155}(1154,·)$, $\chi_{1155}(1091,·)$, $\chi_{1155}(776,·)$, $\chi_{1155}(841,·)$, $\chi_{1155}(524,·)$, $\chi_{1155}(461,·)$, $\chi_{1155}(526,·)$, $\chi_{1155}(169,·)$, $\chi_{1155}(986,·)$, $\chi_{1155}(1114,·)$, $\chi_{1155}(734,·)$, $\chi_{1155}(421,·)$, $\chi_{1155}(41,·)$, $\chi_{1155}(629,·)$, $\chi_{1155}(694,·)$, $\chi_{1155}(631,·)$, $\chi_{1155}(314,·)$, $\chi_{1155}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{35242} a^{12} - \frac{1290}{17621} a^{11} - \frac{1700}{17621} a^{10} - \frac{2551}{35242} a^{9} + \frac{1260}{17621} a^{8} - \frac{3619}{17621} a^{7} + \frac{7669}{35242} a^{6} - \frac{633}{17621} a^{5} - \frac{5037}{17621} a^{4} - \frac{15237}{35242} a^{3} + \frac{11867}{35242} a^{2} - \frac{2857}{35242} a + \frac{6378}{17621}$, $\frac{1}{35242} a^{13} + \frac{7}{263} a^{11} + \frac{707}{35242} a^{10} + \frac{5597}{17621} a^{9} + \frac{4917}{17621} a^{8} + \frac{11889}{35242} a^{7} + \frac{6996}{17621} a^{6} + \frac{576}{17621} a^{5} + \frac{2439}{35242} a^{4} - \frac{4763}{35242} a^{3} - \frac{11295}{35242} a^{2} + \frac{3637}{17621} a - \frac{2774}{17621}$, $\frac{1}{35242} a^{14} + \frac{3335}{17621} a^{11} - \frac{3314}{17621} a^{10} + \frac{3108}{17621} a^{9} - \frac{4139}{17621} a^{8} + \frac{765}{17621} a^{7} - \frac{1501}{17621} a^{6} + \frac{4670}{17621} a^{5} - \frac{207}{35242} a^{4} + \frac{8001}{35242} a^{3} - \frac{5121}{35242} a^{2} + \frac{13547}{35242} a - \frac{7}{526}$, $\frac{1}{35242} a^{15} + \frac{1938}{17621} a^{11} + \frac{5989}{35242} a^{10} + \frac{2627}{35242} a^{9} - \frac{14057}{35242} a^{8} - \frac{3541}{17621} a^{7} - \frac{3374}{17621} a^{6} - \frac{14067}{35242} a^{5} + \frac{6354}{17621} a^{4} + \frac{2681}{17621} a^{3} - \frac{1716}{17621} a^{2} - \frac{10201}{35242} a + \frac{9289}{35242}$, $\frac{1}{35242} a^{16} - \frac{2659}{35242} a^{11} + \frac{519}{35242} a^{10} + \frac{5859}{35242} a^{9} - \frac{6284}{17621} a^{8} - \frac{2446}{17621} a^{7} + \frac{5137}{35242} a^{6} - \frac{7078}{17621} a^{5} + \frac{2025}{17621} a^{4} - \frac{5206}{17621} a^{3} - \frac{15883}{35242} a^{2} + \frac{17033}{35242} a + \frac{1135}{17621}$, $\frac{1}{3136538} a^{17} + \frac{11}{3136538} a^{16} + \frac{15}{3136538} a^{15} - \frac{5}{3136538} a^{14} + \frac{21}{1568269} a^{13} - \frac{21}{3136538} a^{12} + \frac{154469}{3136538} a^{11} - \frac{235115}{1568269} a^{10} - \frac{1212441}{3136538} a^{9} + \frac{404623}{3136538} a^{8} - \frac{78801}{3136538} a^{7} + \frac{686687}{3136538} a^{6} + \frac{766371}{1568269} a^{5} + \frac{770668}{1568269} a^{4} + \frac{448341}{3136538} a^{3} - \frac{1493503}{3136538} a^{2} + \frac{324727}{1568269} a - \frac{705672}{1568269}$, $\frac{1}{3136538} a^{18} - \frac{17}{3136538} a^{16} + \frac{4}{1568269} a^{15} + \frac{4}{1568269} a^{14} - \frac{19}{1568269} a^{13} + \frac{9}{1568269} a^{12} + \frac{415171}{3136538} a^{11} + \frac{126219}{1568269} a^{10} + \frac{648836}{1568269} a^{9} - \frac{384435}{1568269} a^{8} - \frac{1416877}{3136538} a^{7} - \frac{458021}{1568269} a^{6} - \frac{788241}{3136538} a^{5} + \frac{370700}{1568269} a^{4} - \frac{38169}{3136538} a^{3} + \frac{168410}{1568269} a^{2} + \frac{692029}{3136538} a - \frac{970743}{3136538}$, $\frac{1}{158702477615679802416958299942221088106022138952672323498} a^{19} + \frac{8981477594118496816560374472132061712673342385676}{79351238807839901208479149971110544053011069476336161749} a^{18} - \frac{14354745483502657864018744554792829183409997414661}{158702477615679802416958299942221088106022138952672323498} a^{17} + \frac{1119647878040969294075074672033208938460730164996350}{79351238807839901208479149971110544053011069476336161749} a^{16} + \frac{1695996615605411456407821038629965432231919244796669}{158702477615679802416958299942221088106022138952672323498} a^{15} + \frac{1406482650283625192028884929872587842117888244702301}{158702477615679802416958299942221088106022138952672323498} a^{14} - \frac{2991437208509311538099057173701525559317650348961}{6900107722420860974650360867053090787218353867507492326} a^{13} - \frac{1380771581971622434976659008284574266621594167487271}{158702477615679802416958299942221088106022138952672323498} a^{12} - \frac{1429682803400749924025746512046898379377794612619596551}{79351238807839901208479149971110544053011069476336161749} a^{11} + \frac{6046115232142667814582514286653550953386976150823539388}{79351238807839901208479149971110544053011069476336161749} a^{10} + \frac{15627731565039132375216670668294306677628826128272082875}{158702477615679802416958299942221088106022138952672323498} a^{9} - \frac{6315764723847891448181849169651662173861541577247478112}{79351238807839901208479149971110544053011069476336161749} a^{8} + \frac{54880260992156452640619170218581200970636336116217985571}{158702477615679802416958299942221088106022138952672323498} a^{7} - \frac{26481515884567365605107604041577356753619930488938610609}{158702477615679802416958299942221088106022138952672323498} a^{6} - \frac{19857573865458765738680379194112135138066489111534258383}{158702477615679802416958299942221088106022138952672323498} a^{5} + \frac{38030226093254694992463183661554171511107522916703281937}{158702477615679802416958299942221088106022138952672323498} a^{4} + \frac{387887013910926569899979490028414122162184007524670108}{3450053861210430487325180433526545393609176933753746163} a^{3} - \frac{5263530180103340558463531693717801522124726276960247635}{158702477615679802416958299942221088106022138952672323498} a^{2} + \frac{58772686446822236649932883113686353251950658883007675427}{158702477615679802416958299942221088106022138952672323498} a + \frac{19308400380797390405797490504310408960282861875524803144}{79351238807839901208479149971110544053011069476336161749}$

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

Class group and class number

$C_{2}\times C_{22}\times C_{17292}$, which has order $760848$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-231}) \), \(\Q(\sqrt{-1155}) \), \(\Q(\sqrt{5}, \sqrt{-231})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.9630096522760791.1, 10.0.30094051633627471875.1

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7Data 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}$