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Properties

Label 20.12.2828512722...4961.2
Degree $20$
Signature $[12, 4]$
Discriminant $3^{8}\cdot 401^{11}$
Root discriminant $41.94$
Ramified primes $3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_2^4:C_5).C_2$ (as 20T84)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4549, -5984, 41185, 8299, -118818, 110519, 10318, -77650, 50163, -8003, -8075, 10251, -6678, 469, 2046, -864, -125, 133, -11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 11*x^18 + 133*x^17 - 125*x^16 - 864*x^15 + 2046*x^14 + 469*x^13 - 6678*x^12 + 10251*x^11 - 8075*x^10 - 8003*x^9 + 50163*x^8 - 77650*x^7 + 10318*x^6 + 110519*x^5 - 118818*x^4 + 8299*x^3 + 41185*x^2 - 5984*x - 4549)
 
gp: K = bnfinit(x^20 - 6*x^19 - 11*x^18 + 133*x^17 - 125*x^16 - 864*x^15 + 2046*x^14 + 469*x^13 - 6678*x^12 + 10251*x^11 - 8075*x^10 - 8003*x^9 + 50163*x^8 - 77650*x^7 + 10318*x^6 + 110519*x^5 - 118818*x^4 + 8299*x^3 + 41185*x^2 - 5984*x - 4549, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 11 x^{18} + 133 x^{17} - 125 x^{16} - 864 x^{15} + 2046 x^{14} + 469 x^{13} - 6678 x^{12} + 10251 x^{11} - 8075 x^{10} - 8003 x^{9} + 50163 x^{8} - 77650 x^{7} + 10318 x^{6} + 110519 x^{5} - 118818 x^{4} + 8299 x^{3} + 41185 x^{2} - 5984 x - 4549 \)

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:  \(282851272292314208272349925674961=3^{8}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.94$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{93} a^{18} + \frac{14}{93} a^{17} + \frac{7}{93} a^{16} + \frac{5}{31} a^{15} + \frac{5}{31} a^{14} + \frac{1}{93} a^{13} - \frac{4}{93} a^{12} + \frac{34}{93} a^{11} + \frac{41}{93} a^{10} - \frac{38}{93} a^{9} + \frac{5}{31} a^{8} - \frac{41}{93} a^{7} + \frac{29}{93} a^{6} + \frac{43}{93} a^{5} + \frac{5}{31} a^{4} - \frac{19}{93} a^{3} + \frac{4}{93} a^{2} - \frac{4}{93} a + \frac{12}{31}$, $\frac{1}{177281628042252576123764184276697107} a^{19} - \frac{69635052015014570075475867133529}{59093876014084192041254728092232369} a^{18} - \frac{431829593425081662119161055984420}{3110204000741273265329196215380651} a^{17} + \frac{216930140699320706413344112202366}{3110204000741273265329196215380651} a^{16} - \frac{1810644818363120498630790773897080}{177281628042252576123764184276697107} a^{15} - \frac{16213361407672176352489828333375963}{177281628042252576123764184276697107} a^{14} - \frac{27133080787384711499679471225963530}{177281628042252576123764184276697107} a^{13} - \frac{21810578585789840139490992222019058}{177281628042252576123764184276697107} a^{12} + \frac{5125166252150299852031222452604192}{59093876014084192041254728092232369} a^{11} - \frac{18681871651631169239426628875766670}{59093876014084192041254728092232369} a^{10} - \frac{76990766505772623984535144790987473}{177281628042252576123764184276697107} a^{9} - \frac{28490633118343161527844652968160109}{59093876014084192041254728092232369} a^{8} - \frac{49021832010242709079458193084117663}{177281628042252576123764184276697107} a^{7} - \frac{34002327708874558772923269476263393}{177281628042252576123764184276697107} a^{6} - \frac{11849658000447063564035469098390534}{59093876014084192041254728092232369} a^{5} - \frac{4665118242087596153487766398791282}{59093876014084192041254728092232369} a^{4} + \frac{10057374395784803360123503813902827}{59093876014084192041254728092232369} a^{3} - \frac{49737230154548057947116591256875836}{177281628042252576123764184276697107} a^{2} - \frac{48178704578811793414322075199947167}{177281628042252576123764184276697107} a - \frac{10182256021508596859247813066424850}{177281628042252576123764184276697107}$

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:  \( 1120221881.84 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_2^4:C_5).C_2$ (as 20T84):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 20 conjugacy class representatives for $(C_2\times C_2^4:C_5).C_2$
Character table for $(C_2\times C_2^4:C_5).C_2$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed