↑

Properties

Label 20.0.34425542535...000.23
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $94.81$
Ramified primes $2, 3, 5, 11$
Class number $1469152$ (GRH)
Class group $[2, 2, 367288]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48162813931, -29346150362, 42002721191, -28627012626, 23152394787, -12219263196, 6947616282, -2851812468, 1230518499, -397588862, 135278353, -34625998, 9461145, -1902132, 419322, -64188, 11313, -1218, 167, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 167*x^18 - 1218*x^17 + 11313*x^16 - 64188*x^15 + 419322*x^14 - 1902132*x^13 + 9461145*x^12 - 34625998*x^11 + 135278353*x^10 - 397588862*x^9 + 1230518499*x^8 - 2851812468*x^7 + 6947616282*x^6 - 12219263196*x^5 + 23152394787*x^4 - 28627012626*x^3 + 42002721191*x^2 - 29346150362*x + 48162813931)
 
gp: K = bnfinit(x^20 - 10*x^19 + 167*x^18 - 1218*x^17 + 11313*x^16 - 64188*x^15 + 419322*x^14 - 1902132*x^13 + 9461145*x^12 - 34625998*x^11 + 135278353*x^10 - 397588862*x^9 + 1230518499*x^8 - 2851812468*x^7 + 6947616282*x^6 - 12219263196*x^5 + 23152394787*x^4 - 28627012626*x^3 + 42002721191*x^2 - 29346150362*x + 48162813931, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 167 x^{18} - 1218 x^{17} + 11313 x^{16} - 64188 x^{15} + 419322 x^{14} - 1902132 x^{13} + 9461145 x^{12} - 34625998 x^{11} + 135278353 x^{10} - 397588862 x^{9} + 1230518499 x^{8} - 2851812468 x^{7} + 6947616282 x^{6} - 12219263196 x^{5} + 23152394787 x^{4} - 28627012626 x^{3} + 42002721191 x^{2} - 29346150362 x + 48162813931 \)

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:  \(3442554253548220860035959357440000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.81$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(899,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(371,·)$, $\chi_{1320}(529,·)$, $\chi_{1320}(1091,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(491,·)$, $\chi_{1320}(289,·)$, $\chi_{1320}(611,·)$, $\chi_{1320}(131,·)$, $\chi_{1320}(169,·)$, $\chi_{1320}(299,·)$, $\chi_{1320}(49,·)$, $\chi_{1320}(659,·)$, $\chi_{1320}(1139,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(889,·)$, $\chi_{1320}(1019,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{6} + \frac{1}{3} a^{3} - \frac{1}{6}$, $\frac{1}{6} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{36} a^{8} + \frac{1}{18} a^{7} + \frac{1}{18} a^{6} + \frac{1}{18} a^{5} + \frac{1}{36} a^{4} - \frac{1}{18} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9} a - \frac{17}{36}$, $\frac{1}{36} a^{9} - \frac{1}{18} a^{7} - \frac{1}{18} a^{6} - \frac{1}{12} a^{5} + \frac{1}{18} a^{4} + \frac{1}{6} a^{2} + \frac{11}{36} a + \frac{1}{9}$, $\frac{1}{36} a^{10} + \frac{1}{18} a^{7} + \frac{1}{36} a^{6} + \frac{1}{18} a^{4} - \frac{1}{9} a^{3} - \frac{1}{4} a^{2} - \frac{1}{6} a - \frac{5}{18}$, $\frac{1}{36} a^{11} - \frac{1}{12} a^{7} + \frac{1}{18} a^{6} - \frac{1}{18} a^{5} - \frac{5}{36} a^{3} + \frac{2}{9} a^{2} - \frac{1}{2} a - \frac{1}{18}$, $\frac{1}{216} a^{12} - \frac{1}{72} a^{10} - \frac{1}{108} a^{9} + \frac{17}{216} a^{6} - \frac{1}{12} a^{5} - \frac{1}{18} a^{4} + \frac{7}{27} a^{3} + \frac{1}{24} a^{2} - \frac{7}{36} a - \frac{17}{216}$, $\frac{1}{216} a^{13} - \frac{1}{72} a^{11} - \frac{1}{108} a^{10} + \frac{17}{216} a^{7} - \frac{1}{12} a^{6} - \frac{1}{18} a^{5} - \frac{2}{27} a^{4} - \frac{7}{24} a^{3} + \frac{5}{36} a^{2} + \frac{55}{216} a - \frac{1}{3}$, $\frac{1}{216} a^{14} - \frac{1}{108} a^{11} - \frac{1}{72} a^{10} - \frac{1}{216} a^{8} - \frac{1}{12} a^{7} - \frac{1}{72} a^{6} - \frac{2}{27} a^{5} + \frac{5}{72} a^{4} + \frac{5}{36} a^{3} + \frac{25}{54} a^{2} - \frac{4}{9} a + \frac{13}{72}$, $\frac{1}{216} a^{15} - \frac{1}{72} a^{11} + \frac{1}{216} a^{9} - \frac{1}{72} a^{7} + \frac{1}{18} a^{6} - \frac{1}{72} a^{5} + \frac{1}{18} a^{4} + \frac{1}{27} a^{3} + \frac{1}{18} a^{2} - \frac{17}{72} a - \frac{11}{27}$, $\frac{1}{1296} a^{16} - \frac{1}{648} a^{15} + \frac{1}{648} a^{14} + \frac{1}{648} a^{12} + \frac{1}{648} a^{11} - \frac{7}{648} a^{10} - \frac{1}{216} a^{9} + \frac{1}{1296} a^{8} - \frac{1}{216} a^{7} - \frac{7}{648} a^{6} + \frac{17}{648} a^{5} - \frac{1}{81} a^{4} + \frac{25}{108} a^{3} + \frac{26}{81} a^{2} + \frac{157}{648} a + \frac{607}{1296}$, $\frac{1}{55728} a^{17} + \frac{13}{55728} a^{16} - \frac{31}{13932} a^{15} - \frac{1}{3096} a^{14} + \frac{37}{27864} a^{13} + \frac{61}{27864} a^{12} - \frac{143}{13932} a^{11} - \frac{7}{1032} a^{10} + \frac{283}{55728} a^{9} - \frac{197}{18576} a^{8} + \frac{973}{13932} a^{7} - \frac{745}{27864} a^{6} - \frac{1331}{27864} a^{5} + \frac{41}{4644} a^{4} - \frac{481}{3483} a^{3} + \frac{1472}{3483} a^{2} + \frac{7873}{55728} a + \frac{3565}{18576}$, $\frac{1}{174532391250303472425689381136} a^{18} - \frac{1}{19392487916700385825076597904} a^{17} - \frac{1845214995151469316869603}{58177463750101157475229793712} a^{16} + \frac{1230143330100979544579741}{4848121979175096456269149476} a^{15} + \frac{41775765066452035027620841}{29088731875050578737614896856} a^{14} - \frac{732718341437877472991237}{1212030494793774114067287369} a^{13} + \frac{35191877853160836426342653}{29088731875050578737614896856} a^{12} + \frac{353285984527220168685479761}{29088731875050578737614896856} a^{11} + \frac{448571557053639847063659145}{58177463750101157475229793712} a^{10} + \frac{2409608105063518704809325473}{174532391250303472425689381136} a^{9} - \frac{218944846570566545167359689}{58177463750101157475229793712} a^{8} + \frac{174099110764193092654999427}{9696243958350192912538298952} a^{7} - \frac{2333967754044400681958031677}{29088731875050578737614896856} a^{6} - \frac{13570282859948166090205693}{266869099771106226950595384} a^{5} - \frac{482812992895874361020493911}{29088731875050578737614896856} a^{4} - \frac{3429311032980771002704158239}{9696243958350192912538298952} a^{3} + \frac{15621203203091284474638136751}{58177463750101157475229793712} a^{2} - \frac{25730046047897411259699381733}{58177463750101157475229793712} a - \frac{58038645782216457735164403913}{174532391250303472425689381136}$, $\frac{1}{847393680243472176301072874147273328} a^{19} + \frac{1213801}{423696840121736088150536437073636664} a^{18} - \frac{6138739552623857539721314489325}{847393680243472176301072874147273328} a^{17} - \frac{144092841698654952617926894129525}{423696840121736088150536437073636664} a^{16} - \frac{250899484353084053745393806169985}{211848420060868044075268218536818332} a^{15} - \frac{12539508957200122439291428993778}{17654035005072337006272351544734861} a^{14} + \frac{407149620215429619796736045098921}{423696840121736088150536437073636664} a^{13} - \frac{17441043203847129247127951546731}{211848420060868044075268218536818332} a^{12} - \frac{7288158826170128338195608354349067}{847393680243472176301072874147273328} a^{11} - \frac{271946683579450305735213807454655}{423696840121736088150536437073636664} a^{10} + \frac{10164534351926465919386095787845715}{847393680243472176301072874147273328} a^{9} + \frac{271481156161868229042077468985257}{47077426680192898683392937452626296} a^{8} - \frac{8949390710241189095722460480353997}{211848420060868044075268218536818332} a^{7} - \frac{770679964410654539550992180154269}{52962105015217011018817054634204583} a^{6} - \frac{2025736132520789275888339714954559}{105924210030434022037634109268409166} a^{5} + \frac{169786591907528612183194984734595}{7846237780032149780565489575437716} a^{4} + \frac{217358425415279293210757418442269137}{847393680243472176301072874147273328} a^{3} - \frac{187740331816701449296648452057242627}{423696840121736088150536437073636664} a^{2} - \frac{46344417320077873693833626193358789}{282464560081157392100357624715757776} a + \frac{186278378485262462811144908494431107}{423696840121736088150536437073636664}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{367288}$, which has order $1469152$ (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

\(\Q(\sqrt{-66}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-330}) \), \(\Q(\sqrt{5}, \sqrt{-66})\), \(\Q(\zeta_{11})^+\), 10.0.18775450875101184.1, 10.10.669871503125.1, 10.0.58673283984691200000.1

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 R R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
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}$
$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}$