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Properties

Label 20.8.17672461644...8125.1
Degree $20$
Signature $[8, 6]$
Discriminant $5^{14}\cdot 6029^{7}$
Root discriminant $64.92$
Ramified primes $5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![542881, -389432, -193979, 468496, -903700, 1693891, -1840394, 1649663, -1248062, 585829, -121445, -51366, 59993, -21632, 2220, 1068, -612, 109, -1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - x^18 + 109*x^17 - 612*x^16 + 1068*x^15 + 2220*x^14 - 21632*x^13 + 59993*x^12 - 51366*x^11 - 121445*x^10 + 585829*x^9 - 1248062*x^8 + 1649663*x^7 - 1840394*x^6 + 1693891*x^5 - 903700*x^4 + 468496*x^3 - 193979*x^2 - 389432*x + 542881)
 
gp: K = bnfinit(x^20 - 3*x^19 - x^18 + 109*x^17 - 612*x^16 + 1068*x^15 + 2220*x^14 - 21632*x^13 + 59993*x^12 - 51366*x^11 - 121445*x^10 + 585829*x^9 - 1248062*x^8 + 1649663*x^7 - 1840394*x^6 + 1693891*x^5 - 903700*x^4 + 468496*x^3 - 193979*x^2 - 389432*x + 542881, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - x^{18} + 109 x^{17} - 612 x^{16} + 1068 x^{15} + 2220 x^{14} - 21632 x^{13} + 59993 x^{12} - 51366 x^{11} - 121445 x^{10} + 585829 x^{9} - 1248062 x^{8} + 1649663 x^{7} - 1840394 x^{6} + 1693891 x^{5} - 903700 x^{4} + 468496 x^{3} - 193979 x^{2} - 389432 x + 542881 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1767246164450779958943837335205078125=5^{14}\cdot 6029^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.92$
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:  $5, 6029$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{15} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{15} - \frac{1}{5} a^{14} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5943245668465143442213505463925469993325500005201876455620218645} a^{19} + \frac{194234116309521958080412014174054586874556018204432182667551769}{5943245668465143442213505463925469993325500005201876455620218645} a^{18} - \frac{37728562555721407799649645929313528642194712481160142962711713}{1188649133693028688442701092785093998665100001040375291124043729} a^{17} - \frac{126664005605943413160180265246348322980200134338709823681837526}{5943245668465143442213505463925469993325500005201876455620218645} a^{16} + \frac{510066618550608045171745214780092860576438312049062234016093189}{5943245668465143442213505463925469993325500005201876455620218645} a^{15} - \frac{17176631127616072274808601005888920113012498448052244121709369}{204939505809142877317707084962947241149155172593168153642076505} a^{14} + \frac{2382782743483805588698032604961103239816488584536057519902609343}{5943245668465143442213505463925469993325500005201876455620218645} a^{13} + \frac{2959075967830104869110770229331207484236360646589127367742544503}{5943245668465143442213505463925469993325500005201876455620218645} a^{12} + \frac{2700618449608995696901207446034844796778762576849398618573948069}{5943245668465143442213505463925469993325500005201876455620218645} a^{11} + \frac{1766956563074840715635916893650170550199149524208088584044584263}{5943245668465143442213505463925469993325500005201876455620218645} a^{10} - \frac{809242975231794339503659100516166706583534236088803778852016388}{5943245668465143442213505463925469993325500005201876455620218645} a^{9} - \frac{162646010975482386479420713492626080427221915223946205671625102}{5943245668465143442213505463925469993325500005201876455620218645} a^{8} - \frac{515989925394569706309253974720765622025945435088779202712898512}{1188649133693028688442701092785093998665100001040375291124043729} a^{7} - \frac{42535621646967413120571210341095527260659036338030014079286673}{204939505809142877317707084962947241149155172593168153642076505} a^{6} - \frac{918626053649550286313690245717466233525060584480588660094008042}{5943245668465143442213505463925469993325500005201876455620218645} a^{5} - \frac{2377523168532704160032261808755101558603637469273796035431899572}{5943245668465143442213505463925469993325500005201876455620218645} a^{4} - \frac{2348300449182505461684032639106827460393112390306589884736696259}{5943245668465143442213505463925469993325500005201876455620218645} a^{3} - \frac{157463160458877094743451146742106461552338689645955650398729303}{457172743728087957093346574148113076409653846553990496586170665} a^{2} - \frac{1989503012585286245118047144931348622780080812601660292872932162}{5943245668465143442213505463925469993325500005201876455620218645} a + \frac{1488707026267824769740514631467828543482071930979971908350323051}{5943245668465143442213505463925469993325500005201876455620218645}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 30722930196.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed