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Properties

Label 20.20.5113174438...4704.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 3^{10}\cdot 41^{2}\cdot 53^{12}$
Root discriminant $54.38$
Ramified primes $2, 3, 41, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T196

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-72, -36, 1713, 2514, -10970, -23874, 15374, 60464, 5235, -63636, -22753, 31644, 15963, -7524, -4732, 816, 666, -32, -43, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 43*x^18 - 32*x^17 + 666*x^16 + 816*x^15 - 4732*x^14 - 7524*x^13 + 15963*x^12 + 31644*x^11 - 22753*x^10 - 63636*x^9 + 5235*x^8 + 60464*x^7 + 15374*x^6 - 23874*x^5 - 10970*x^4 + 2514*x^3 + 1713*x^2 - 36*x - 72)
 
gp: K = bnfinit(x^20 - 43*x^18 - 32*x^17 + 666*x^16 + 816*x^15 - 4732*x^14 - 7524*x^13 + 15963*x^12 + 31644*x^11 - 22753*x^10 - 63636*x^9 + 5235*x^8 + 60464*x^7 + 15374*x^6 - 23874*x^5 - 10970*x^4 + 2514*x^3 + 1713*x^2 - 36*x - 72, 1)
 

Normalized defining polynomial

\( x^{20} - 43 x^{18} - 32 x^{17} + 666 x^{16} + 816 x^{15} - 4732 x^{14} - 7524 x^{13} + 15963 x^{12} + 31644 x^{11} - 22753 x^{10} - 63636 x^{9} + 5235 x^{8} + 60464 x^{7} + 15374 x^{6} - 23874 x^{5} - 10970 x^{4} + 2514 x^{3} + 1713 x^{2} - 36 x - 72 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(51131744381789867152791425283784704=2^{20}\cdot 3^{10}\cdot 41^{2}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.38$
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:  $2, 3, 41, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{207} a^{16} + \frac{14}{207} a^{15} - \frac{20}{207} a^{14} + \frac{20}{207} a^{13} + \frac{62}{207} a^{12} - \frac{31}{69} a^{11} + \frac{10}{207} a^{10} - \frac{1}{207} a^{9} - \frac{2}{9} a^{8} - \frac{8}{69} a^{7} + \frac{28}{207} a^{6} - \frac{2}{9} a^{5} + \frac{98}{207} a^{4} - \frac{13}{207} a^{3} - \frac{85}{207} a^{2} - \frac{4}{69} a - \frac{14}{69}$, $\frac{1}{207} a^{17} - \frac{1}{23} a^{15} + \frac{8}{69} a^{14} - \frac{80}{207} a^{13} + \frac{5}{207} a^{12} - \frac{68}{207} a^{11} - \frac{1}{69} a^{10} - \frac{101}{207} a^{9} - \frac{70}{207} a^{8} + \frac{19}{207} a^{7} - \frac{31}{69} a^{6} + \frac{52}{207} a^{5} - \frac{5}{207} a^{4} - \frac{41}{207} a^{3} + \frac{74}{207} a^{2} - \frac{9}{23} a - \frac{11}{69}$, $\frac{1}{207} a^{18} + \frac{4}{69} a^{15} + \frac{16}{207} a^{14} + \frac{47}{207} a^{13} + \frac{7}{207} a^{12} + \frac{19}{69} a^{11} + \frac{58}{207} a^{10} - \frac{10}{207} a^{9} - \frac{50}{207} a^{8} - \frac{11}{69} a^{7} - \frac{41}{207} a^{6} + \frac{64}{207} a^{5} - \frac{56}{207} a^{4} + \frac{26}{207} a^{3} - \frac{2}{23} a^{2} + \frac{22}{69} a + \frac{4}{23}$, $\frac{1}{668676144757479071922} a^{19} - \frac{476062562657044817}{334338072378739535961} a^{18} - \frac{16900391042210891}{29072875859020829214} a^{17} - \frac{12676125727585904}{14536437929510414607} a^{16} - \frac{1889665159092000077}{111446024126246511987} a^{15} - \frac{32502990407867354585}{334338072378739535961} a^{14} + \frac{17797309000723461185}{37148674708748837329} a^{13} + \frac{161312045561205503281}{334338072378739535961} a^{12} - \frac{16455750204376157769}{74297349417497674658} a^{11} + \frac{119766747095277997945}{334338072378739535961} a^{10} + \frac{13093247518445951957}{74297349417497674658} a^{9} - \frac{162998494073909624159}{334338072378739535961} a^{8} + \frac{17723978480616558353}{74297349417497674658} a^{7} - \frac{24441589678744774984}{334338072378739535961} a^{6} - \frac{135255702024681189400}{334338072378739535961} a^{5} + \frac{116447956068809255545}{334338072378739535961} a^{4} - \frac{21149075858321068211}{334338072378739535961} a^{3} - \frac{30171318300526076414}{111446024126246511987} a^{2} + \frac{91154428605688472935}{222892048252493023974} a + \frac{17757074498631785538}{37148674708748837329}$

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:  $19$
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:  \( 343897225324 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T196:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1280
The 44 conjugacy class representatives for t20n196
Character table for t20n196 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 5.5.2382032.1, 10.10.18843607887208704.1, 10.10.2791645612919808.1, 10.10.5515202308451328.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$