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Properties

Label 20.12.1806627930...5625.1
Degree $20$
Signature $[12, 4]$
Discriminant $3^{16}\cdot 5^{10}\cdot 73^{10}$
Root discriminant $46.01$
Ramified primes $3, 5, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T277

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-751, 71, 57265, -485, -171224, 76952, 176120, -131413, -84323, 90045, 18864, -32469, -172, 6806, -962, -857, 220, 56, -22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 22*x^18 + 56*x^17 + 220*x^16 - 857*x^15 - 962*x^14 + 6806*x^13 - 172*x^12 - 32469*x^11 + 18864*x^10 + 90045*x^9 - 84323*x^8 - 131413*x^7 + 176120*x^6 + 76952*x^5 - 171224*x^4 - 485*x^3 + 57265*x^2 + 71*x - 751)
 
gp: K = bnfinit(x^20 - x^19 - 22*x^18 + 56*x^17 + 220*x^16 - 857*x^15 - 962*x^14 + 6806*x^13 - 172*x^12 - 32469*x^11 + 18864*x^10 + 90045*x^9 - 84323*x^8 - 131413*x^7 + 176120*x^6 + 76952*x^5 - 171224*x^4 - 485*x^3 + 57265*x^2 + 71*x - 751, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 22 x^{18} + 56 x^{17} + 220 x^{16} - 857 x^{15} - 962 x^{14} + 6806 x^{13} - 172 x^{12} - 32469 x^{11} + 18864 x^{10} + 90045 x^{9} - 84323 x^{8} - 131413 x^{7} + 176120 x^{6} + 76952 x^{5} - 171224 x^{4} - 485 x^{3} + 57265 x^{2} + 71 x - 751 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1806627930211353113514833291015625=3^{16}\cdot 5^{10}\cdot 73^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.01$
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:  $3, 5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{12} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{9} a^{3} - \frac{2}{9}$, $\frac{1}{117} a^{16} + \frac{2}{117} a^{15} + \frac{2}{39} a^{14} - \frac{11}{117} a^{13} + \frac{5}{117} a^{12} + \frac{2}{39} a^{11} + \frac{6}{13} a^{10} + \frac{4}{39} a^{9} - \frac{5}{39} a^{8} - \frac{2}{39} a^{7} - \frac{19}{39} a^{6} + \frac{2}{13} a^{5} - \frac{20}{117} a^{4} + \frac{8}{117} a^{3} - \frac{19}{39} a^{2} - \frac{17}{117} a - \frac{22}{117}$, $\frac{1}{351} a^{17} - \frac{1}{351} a^{16} - \frac{1}{27} a^{15} + \frac{49}{351} a^{14} - \frac{40}{351} a^{13} + \frac{17}{351} a^{12} - \frac{3}{13} a^{11} + \frac{41}{117} a^{10} + \frac{35}{117} a^{9} - \frac{2}{9} a^{8} + \frac{4}{9} a^{7} + \frac{37}{117} a^{6} + \frac{4}{351} a^{5} + \frac{29}{351} a^{4} - \frac{94}{351} a^{3} - \frac{80}{351} a^{2} - \frac{127}{351} a - \frac{25}{351}$, $\frac{1}{351} a^{18} + \frac{1}{351} a^{16} - \frac{4}{117} a^{15} - \frac{2}{39} a^{14} + \frac{46}{351} a^{13} + \frac{50}{351} a^{12} + \frac{5}{117} a^{11} - \frac{5}{117} a^{10} + \frac{10}{39} a^{9} - \frac{10}{117} a^{8} + \frac{20}{117} a^{7} + \frac{79}{351} a^{6} + \frac{23}{117} a^{5} - \frac{14}{351} a^{4} + \frac{34}{117} a^{3} - \frac{14}{39} a^{2} + \frac{61}{351} a + \frac{152}{351}$, $\frac{1}{92104350365937067855585086863691} a^{19} + \frac{20018782567242428925051815254}{92104350365937067855585086863691} a^{18} - \frac{126219092771222343962418124943}{92104350365937067855585086863691} a^{17} + \frac{228773368336925020630282354477}{92104350365937067855585086863691} a^{16} - \frac{766684626418365989862642136111}{30701450121979022618528362287897} a^{15} + \frac{1023498597333776245602268466653}{7084950028149005219660391297207} a^{14} + \frac{53346597190243448094740898850}{787216669794333913295599033023} a^{13} - \frac{1278842126082036995864844836734}{92104350365937067855585086863691} a^{12} - \frac{334970588980957364569034005829}{1137090745258482319204754158811} a^{11} + \frac{159548749736843867522205393883}{829768922215649259960226007781} a^{10} - \frac{3605468525493980470159898677015}{30701450121979022618528362287897} a^{9} + \frac{14143282692887888471106584565490}{30701450121979022618528362287897} a^{8} - \frac{33867256472521073508548559251840}{92104350365937067855585086863691} a^{7} + \frac{14111629155985454008398880278382}{92104350365937067855585086863691} a^{6} + \frac{4568533987929204410324062177123}{92104350365937067855585086863691} a^{5} - \frac{8278313968956618211065898958882}{92104350365937067855585086863691} a^{4} + \frac{1884370500920954122480932174370}{30701450121979022618528362287897} a^{3} - \frac{7904719667118038060817700480523}{92104350365937067855585086863691} a^{2} - \frac{315200225155174722169658617918}{10233816707326340872842787429299} a + \frac{36597049725486784596213852369128}{92104350365937067855585086863691}$

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:  $15$
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:  \( 3441909364.82 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T277:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 3840
The 48 conjugacy class representatives for t20n277
Character table for t20n277 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.10791225.1, 10.6.42504446005228125.1, 10.10.582252685003125.1, 10.6.8500889201045625.2

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

Sibling fields

Degree 20 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
73.6.4.2$x^{6} - 73 x^{3} + 58619$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$