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Properties

Label 20.0.96976735766...0625.3
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 11^{16}\cdot 43^{10}$
Root discriminant $99.85$
Ramified primes $5, 11, 43$
Class number $2464000$ (GRH)
Class group $[4, 20, 20, 1540]$ (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![169930201079, -87034245171, 115845571171, -47487519227, 34734256499, -11793282781, 6169036807, -1774890291, 733164737, -181154943, 61975866, -13199759, 3828520, -697337, 172889, -26230, 5514, -648, 111, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 111*x^18 - 648*x^17 + 5514*x^16 - 26230*x^15 + 172889*x^14 - 697337*x^13 + 3828520*x^12 - 13199759*x^11 + 61975866*x^10 - 181154943*x^9 + 733164737*x^8 - 1774890291*x^7 + 6169036807*x^6 - 11793282781*x^5 + 34734256499*x^4 - 47487519227*x^3 + 115845571171*x^2 - 87034245171*x + 169930201079)
 
gp: K = bnfinit(x^20 - 8*x^19 + 111*x^18 - 648*x^17 + 5514*x^16 - 26230*x^15 + 172889*x^14 - 697337*x^13 + 3828520*x^12 - 13199759*x^11 + 61975866*x^10 - 181154943*x^9 + 733164737*x^8 - 1774890291*x^7 + 6169036807*x^6 - 11793282781*x^5 + 34734256499*x^4 - 47487519227*x^3 + 115845571171*x^2 - 87034245171*x + 169930201079, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 111 x^{18} - 648 x^{17} + 5514 x^{16} - 26230 x^{15} + 172889 x^{14} - 697337 x^{13} + 3828520 x^{12} - 13199759 x^{11} + 61975866 x^{10} - 181154943 x^{9} + 733164737 x^{8} - 1774890291 x^{7} + 6169036807 x^{6} - 11793282781 x^{5} + 34734256499 x^{4} - 47487519227 x^{3} + 115845571171 x^{2} - 87034245171 x + 169930201079 \)

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:  \(9697673576628699464987098790938369140625=5^{10}\cdot 11^{16}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.85$
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:  $5, 11, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2365=5\cdot 11\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2365}(1,·)$, $\chi_{2365}(386,·)$, $\chi_{2365}(1291,·)$, $\chi_{2365}(1676,·)$, $\chi_{2365}(1549,·)$, $\chi_{2365}(1934,·)$, $\chi_{2365}(2194,·)$, $\chi_{2365}(214,·)$, $\chi_{2365}(474,·)$, $\chi_{2365}(859,·)$, $\chi_{2365}(861,·)$, $\chi_{2365}(1246,·)$, $\chi_{2365}(1764,·)$, $\chi_{2365}(2149,·)$, $\chi_{2365}(1076,·)$, $\chi_{2365}(1461,·)$, $\chi_{2365}(1334,·)$, $\chi_{2365}(1719,·)$, $\chi_{2365}(1721,·)$, $\chi_{2365}(2106,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{36079} a^{18} - \frac{6167}{36079} a^{17} + \frac{17114}{36079} a^{16} + \frac{11863}{36079} a^{15} - \frac{3829}{36079} a^{14} - \frac{17245}{36079} a^{13} - \frac{5241}{36079} a^{12} + \frac{14858}{36079} a^{11} - \frac{15253}{36079} a^{10} - \frac{8870}{36079} a^{9} + \frac{17988}{36079} a^{8} + \frac{13980}{36079} a^{7} + \frac{17709}{36079} a^{6} + \frac{9674}{36079} a^{5} + \frac{4673}{36079} a^{4} - \frac{16487}{36079} a^{3} - \frac{6544}{36079} a^{2} + \frac{13882}{36079} a - \frac{14088}{36079}$, $\frac{1}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{19} - \frac{174010145922185799880080301323296166387338309099148756715565427737894506}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{18} - \frac{7184683742095273441485496688409095157853276842757941545665702209964557010706}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{17} - \frac{2785432460404079796808599331504374440092352934364175129977483046978858294577}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{16} - \frac{1565928885420644903850037083879588499127867072363594482737123115024352115016}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{15} + \frac{11290671714942851988532422767555739083072879127939999809610222070733497106056}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{14} + \frac{920714981158648188392086470551902734947734361493250271980680947157690815842}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{13} + \frac{10116234389290435196996128617387358523454192414447254793921672313139703088668}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{12} + \frac{7204431045045330771751503925252539549798368371037843780968159209622131046357}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{11} - \frac{6845396770658040103117565866373679720247736093859808472984530637075489562453}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{10} - \frac{8520461160286636939993521078647520485046193981933720831849873110174890150699}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{9} - \frac{4172864775086954515620586519406322779229239807925844303251720625666048522350}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{8} + \frac{6748323694311012203359456333855596667336715789991785668706235025472849109649}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{7} - \frac{8052494355197303039461412795265539923050777780029449561936919571935908451182}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{6} + \frac{9313840213206032981063553320015677245547324115659017767227499030255573659681}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{5} - \frac{2063596308287871269386230243712237164854524669768261491198300843144766443631}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{4} - \frac{5091312403171789109313741318417364997310589495871786806533928791899411452354}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{3} - \frac{870563720152405037113937330609225582663285132444624251214424348936639264875}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a^{2} + \frac{1506696755762180871320588990179495116800689269901468005903445427763423570945}{22635296669342126867814761245642887320987241137811174360527415094745287572857} a - \frac{1647008345973003572896012673608033391823266347761923538045131796794396512233}{22635296669342126867814761245642887320987241137811174360527415094745287572857}$

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

Class group and class number

$C_{4}\times C_{20}\times C_{20}\times C_{1540}$, which has order $2464000$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{-215}) \), \(\Q(\sqrt{5}, \sqrt{-43})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.31512565339032283.3, 10.0.98476766684475884375.2

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 ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$43$43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$