↑

Properties

Label 20.0.13812136416...5369.3
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{18}\cdot 29^{10}$
Root discriminant $80.73$
Ramified primes $3, 11, 29$
Class number $503250$ (GRH)
Class group $[5, 100650]$ (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![5697974689, 2645057470, 1174385135, 1900714915, 2079671110, 233938551, 1131385739, -28193649, 303154852, -11288717, 45616682, -1544654, 4114474, -116810, 226458, -4984, 7426, -111, 133, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 133*x^18 - 111*x^17 + 7426*x^16 - 4984*x^15 + 226458*x^14 - 116810*x^13 + 4114474*x^12 - 1544654*x^11 + 45616682*x^10 - 11288717*x^9 + 303154852*x^8 - 28193649*x^7 + 1131385739*x^6 + 233938551*x^5 + 2079671110*x^4 + 1900714915*x^3 + 1174385135*x^2 + 2645057470*x + 5697974689)
 
gp: K = bnfinit(x^20 - x^19 + 133*x^18 - 111*x^17 + 7426*x^16 - 4984*x^15 + 226458*x^14 - 116810*x^13 + 4114474*x^12 - 1544654*x^11 + 45616682*x^10 - 11288717*x^9 + 303154852*x^8 - 28193649*x^7 + 1131385739*x^6 + 233938551*x^5 + 2079671110*x^4 + 1900714915*x^3 + 1174385135*x^2 + 2645057470*x + 5697974689, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 133 x^{18} - 111 x^{17} + 7426 x^{16} - 4984 x^{15} + 226458 x^{14} - 116810 x^{13} + 4114474 x^{12} - 1544654 x^{11} + 45616682 x^{10} - 11288717 x^{9} + 303154852 x^{8} - 28193649 x^{7} + 1131385739 x^{6} + 233938551 x^{5} + 2079671110 x^{4} + 1900714915 x^{3} + 1174385135 x^{2} + 2645057470 x + 5697974689 \)

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:  \(138121364163981500584415263383168735369=3^{10}\cdot 11^{18}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.73$
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:  $3, 11, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(957=3\cdot 11\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{957}(1,·)$, $\chi_{957}(898,·)$, $\chi_{957}(262,·)$, $\chi_{957}(28,·)$, $\chi_{957}(521,·)$, $\chi_{957}(842,·)$, $\chi_{957}(782,·)$, $\chi_{957}(784,·)$, $\chi_{957}(86,·)$, $\chi_{957}(668,·)$, $\chi_{957}(608,·)$, $\chi_{957}(610,·)$, $\chi_{957}(233,·)$, $\chi_{957}(811,·)$, $\chi_{957}(494,·)$, $\chi_{957}(434,·)$, $\chi_{957}(755,·)$, $\chi_{957}(376,·)$, $\chi_{957}(697,·)$, $\chi_{957}(637,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{66263} a^{11} + \frac{77}{66263} a^{9} + \frac{2156}{66263} a^{7} + \frac{26411}{66263} a^{5} - \frac{471}{66263} a^{3} - \frac{13912}{66263} a + \frac{7131}{66263}$, $\frac{1}{66263} a^{12} + \frac{77}{66263} a^{10} + \frac{2156}{66263} a^{8} + \frac{26411}{66263} a^{6} - \frac{471}{66263} a^{4} - \frac{13912}{66263} a^{2} + \frac{7131}{66263} a$, $\frac{1}{66263} a^{13} - \frac{3773}{66263} a^{9} - \frac{7075}{66263} a^{7} + \frac{20035}{66263} a^{5} + \frac{22355}{66263} a^{3} + \frac{7131}{66263} a^{2} + \frac{11016}{66263} a - \frac{18983}{66263}$, $\frac{1}{66263} a^{14} - \frac{3773}{66263} a^{10} - \frac{7075}{66263} a^{8} + \frac{20035}{66263} a^{6} + \frac{22355}{66263} a^{4} + \frac{7131}{66263} a^{3} + \frac{11016}{66263} a^{2} - \frac{18983}{66263} a$, $\frac{1}{530104} a^{15} + \frac{1}{530104} a^{13} - \frac{1}{530104} a^{12} + \frac{1}{265052} a^{11} - \frac{16585}{132526} a^{10} + \frac{40519}{265052} a^{9} + \frac{65185}{265052} a^{8} - \frac{131015}{530104} a^{7} - \frac{158937}{530104} a^{6} + \frac{216889}{530104} a^{5} - \frac{191187}{530104} a^{4} + \frac{44447}{530104} a^{3} + \frac{68323}{530104} a^{2} - \frac{166145}{530104} a - \frac{4675}{12328}$, $\frac{1}{8044380526035736} a^{16} - \frac{365537392}{1005547565754467} a^{15} - \frac{39604260327}{8044380526035736} a^{14} + \frac{37277520839}{8044380526035736} a^{13} - \frac{22178594515}{4022190263017868} a^{12} + \frac{14466676825}{2011095131508934} a^{11} - \frac{1042507753247289}{4022190263017868} a^{10} + \frac{747092521400045}{4022190263017868} a^{9} + \frac{1320677273343481}{8044380526035736} a^{8} + \frac{3405049297781119}{8044380526035736} a^{7} + \frac{1593977995890561}{8044380526035736} a^{6} + \frac{3745973893320341}{8044380526035736} a^{5} + \frac{649778614826511}{8044380526035736} a^{4} + \frac{1600763366709547}{8044380526035736} a^{3} - \frac{2878210115383033}{8044380526035736} a^{2} - \frac{207009732149561}{8044380526035736} a - \frac{46414078485769}{1005547565754467}$, $\frac{1}{8044380526035736} a^{17} + \frac{2253195709}{4022190263017868} a^{15} + \frac{19873438919}{8044380526035736} a^{14} - \frac{29827938581}{8044380526035736} a^{13} - \frac{10361032501}{8044380526035736} a^{12} + \frac{3956933055}{1005547565754467} a^{11} + \frac{1383318687227847}{4022190263017868} a^{10} - \frac{2449482356864041}{8044380526035736} a^{9} + \frac{171200701956847}{349755675045032} a^{8} + \frac{823150118692273}{4022190263017868} a^{7} - \frac{234075531453665}{2011095131508934} a^{6} + \frac{292926864224924}{1005547565754467} a^{5} + \frac{427470899499441}{1005547565754467} a^{4} - \frac{1557919838963873}{4022190263017868} a^{3} + \frac{353353681764269}{4022190263017868} a^{2} + \frac{796312153045943}{8044380526035736} a + \frac{154076374299591}{8044380526035736}$, $\frac{1}{8044380526035736} a^{18} - \frac{1368638383}{4022190263017868} a^{15} + \frac{12704431265}{8044380526035736} a^{14} + \frac{3928128312}{1005547565754467} a^{13} + \frac{50094976273}{8044380526035736} a^{12} - \frac{6589562513}{2011095131508934} a^{11} + \frac{2528634314933807}{8044380526035736} a^{10} - \frac{1132676365991961}{8044380526035736} a^{9} + \frac{1839025037434295}{4022190263017868} a^{8} - \frac{3161072712934751}{8044380526035736} a^{7} + \frac{37719771206585}{187078616884552} a^{6} - \frac{173077938252215}{8044380526035736} a^{5} + \frac{3600378921822487}{8044380526035736} a^{4} - \frac{3087600826588047}{8044380526035736} a^{3} + \frac{1739830858399433}{4022190263017868} a^{2} - \frac{1662912471164089}{4022190263017868} a + \frac{3145616257054325}{8044380526035736}$, $\frac{1}{8044380526035736} a^{19} - \frac{7521201495}{8044380526035736} a^{15} + \frac{375347607}{4022190263017868} a^{14} - \frac{810225351}{120065380985608} a^{13} - \frac{3616401694}{1005547565754467} a^{12} + \frac{8504821911}{8044380526035736} a^{11} + \frac{1044823565993619}{8044380526035736} a^{10} - \frac{1396726305299663}{4022190263017868} a^{9} - \frac{1583554094533009}{8044380526035736} a^{8} + \frac{105893685674861}{8044380526035736} a^{7} + \frac{2505107077392351}{8044380526035736} a^{6} - \frac{2516039306086771}{8044380526035736} a^{5} - \frac{124582086399715}{349755675045032} a^{4} - \frac{19893898925827}{87438918761258} a^{3} + \frac{402565356683382}{1005547565754467} a^{2} + \frac{2615764307673847}{8044380526035736} a - \frac{217617811979943}{1005547565754467}$

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

Class group and class number

$C_{5}\times C_{100650}$, which has order $503250$ (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:  \( 125582.779517 \) (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

\(\Q(\sqrt{-319}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{33}, \sqrt{-87})\), \(\Q(\zeta_{11})^+\), 10.0.48364216424306959.3, \(\Q(\zeta_{33})^+\), 10.0.1068409508282417367.3

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data 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}$
$29$29.10.5.2$x^{10} - 707281 x^{2} + 225622639$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
29.10.5.2$x^{10} - 707281 x^{2} + 225622639$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$