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Properties

Label 20.8.66330517200...4976.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{48}\cdot 31^{6}\cdot 227^{4}$
Root discriminant $43.76$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-356, 2192, -3544, -2904, 12252, -6376, -3608, -4600, 9798, 344, -1954, -2776, 819, 772, 52, -112, -47, 64, -12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 12*x^18 + 64*x^17 - 47*x^16 - 112*x^15 + 52*x^14 + 772*x^13 + 819*x^12 - 2776*x^11 - 1954*x^10 + 344*x^9 + 9798*x^8 - 4600*x^7 - 3608*x^6 - 6376*x^5 + 12252*x^4 - 2904*x^3 - 3544*x^2 + 2192*x - 356)
 
gp: K = bnfinit(x^20 - 4*x^19 - 12*x^18 + 64*x^17 - 47*x^16 - 112*x^15 + 52*x^14 + 772*x^13 + 819*x^12 - 2776*x^11 - 1954*x^10 + 344*x^9 + 9798*x^8 - 4600*x^7 - 3608*x^6 - 6376*x^5 + 12252*x^4 - 2904*x^3 - 3544*x^2 + 2192*x - 356, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 12 x^{18} + 64 x^{17} - 47 x^{16} - 112 x^{15} + 52 x^{14} + 772 x^{13} + 819 x^{12} - 2776 x^{11} - 1954 x^{10} + 344 x^{9} + 9798 x^{8} - 4600 x^{7} - 3608 x^{6} - 6376 x^{5} + 12252 x^{4} - 2904 x^{3} - 3544 x^{2} + 2192 x - 356 \)

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:  \(663305172009703483445153131134976=2^{48}\cdot 31^{6}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.76$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{1048179271868089650708370528244} a^{19} + \frac{113340308947458784565866157617}{1048179271868089650708370528244} a^{18} - \frac{59157909825914418556501886957}{1048179271868089650708370528244} a^{17} - \frac{37302785207330731140575285905}{524089635934044825354185264122} a^{16} - \frac{110413855977635718599264184253}{524089635934044825354185264122} a^{15} - \frac{105977738619800813558039146401}{524089635934044825354185264122} a^{14} + \frac{185985749672291121595147129779}{1048179271868089650708370528244} a^{13} - \frac{237874523302581973873811119085}{1048179271868089650708370528244} a^{12} + \frac{35258138556014305530464988869}{1048179271868089650708370528244} a^{11} + \frac{55485424961569952094176634998}{262044817967022412677092632061} a^{10} + \frac{48402130318617123081937445505}{262044817967022412677092632061} a^{9} + \frac{100893503676852255577163488955}{262044817967022412677092632061} a^{8} + \frac{201423534950230179410786678609}{524089635934044825354185264122} a^{7} - \frac{128829101505838593300053401805}{524089635934044825354185264122} a^{6} - \frac{63228650321662332113672672973}{524089635934044825354185264122} a^{5} - \frac{58152486773219782551457026806}{262044817967022412677092632061} a^{4} + \frac{125638077452037785961642557677}{262044817967022412677092632061} a^{3} - \frac{29210523091045195855935285122}{262044817967022412677092632061} a^{2} - \frac{38376793208467987845159760714}{262044817967022412677092632061} a + \frac{83962588970093177415274091788}{262044817967022412677092632061}$

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:  \( 2085443653.74 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.12.26.27$x^{12} - 2 x^{10} + 4 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{3} + 2$$12$$1$$26$12T48$[4/3, 4/3, 2, 3]_{3}^{2}$
31Data not computed
227Data not computed