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Properties

Label 20.10.6500641709...0176.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{10}\cdot 3^{10}\cdot 401^{10}$
Root discriminant $49.05$
Ramified primes $2, 3, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5:D_4$ (as 20T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-59049, 0, -216513, 0, -131220, 0, 242757, 0, 236925, 0, -7533, 0, -26325, 0, 2997, 0, 180, 0, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 33*x^18 + 180*x^16 + 2997*x^14 - 26325*x^12 - 7533*x^10 + 236925*x^8 + 242757*x^6 - 131220*x^4 - 216513*x^2 - 59049)
 
gp: K = bnfinit(x^20 - 33*x^18 + 180*x^16 + 2997*x^14 - 26325*x^12 - 7533*x^10 + 236925*x^8 + 242757*x^6 - 131220*x^4 - 216513*x^2 - 59049, 1)
 

Normalized defining polynomial

\( x^{20} - 33 x^{18} + 180 x^{16} + 2997 x^{14} - 26325 x^{12} - 7533 x^{10} + 236925 x^{8} + 242757 x^{6} - 131220 x^{4} - 216513 x^{2} - 59049 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6500641709341565444982486072370176=-\,2^{10}\cdot 3^{10}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.05$
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:  $2, 3, 401$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{243} a^{8} + \frac{1}{3}$, $\frac{1}{243} a^{9} + \frac{1}{3} a$, $\frac{1}{729} a^{10} + \frac{1}{9} a^{2}$, $\frac{1}{729} a^{11} + \frac{1}{9} a^{3}$, $\frac{1}{2187} a^{12} + \frac{1}{27} a^{4}$, $\frac{1}{2187} a^{13} + \frac{1}{27} a^{5}$, $\frac{1}{6561} a^{14} + \frac{1}{81} a^{6}$, $\frac{1}{13122} a^{15} - \frac{1}{13122} a^{14} - \frac{1}{4374} a^{13} - \frac{1}{4374} a^{12} - \frac{1}{1458} a^{11} - \frac{1}{81} a^{7} - \frac{1}{162} a^{6} + \frac{1}{27} a^{5} - \frac{1}{54} a^{4} - \frac{1}{18} a^{3} - \frac{1}{2}$, $\frac{1}{1535274} a^{16} - \frac{4}{85293} a^{14} - \frac{1}{9477} a^{12} - \frac{1}{1458} a^{11} - \frac{5}{9477} a^{10} + \frac{4}{9477} a^{8} - \frac{1}{54} a^{7} - \frac{29}{2106} a^{6} - \frac{1}{18} a^{5} - \frac{1}{117} a^{4} - \frac{1}{18} a^{3} + \frac{4}{117} a^{2} - \frac{1}{2} a + \frac{79}{234}$, $\frac{1}{1535274} a^{17} + \frac{5}{170586} a^{15} - \frac{1}{13122} a^{14} + \frac{7}{56862} a^{13} + \frac{1}{6318} a^{11} + \frac{4}{9477} a^{9} - \frac{1}{486} a^{8} + \frac{23}{2106} a^{7} + \frac{1}{81} a^{6} - \frac{16}{351} a^{5} + \frac{7}{78} a^{3} - \frac{1}{6} a^{2} + \frac{79}{234} a - \frac{1}{6}$, $\frac{1}{170415414} a^{18} - \frac{5}{56805138} a^{16} - \frac{14}{3155841} a^{14} - \frac{1}{4374} a^{13} - \frac{10}{1051947} a^{12} - \frac{1}{1458} a^{11} - \frac{398}{1051947} a^{10} - \frac{1}{486} a^{9} + \frac{23}{53946} a^{8} + \frac{509}{77922} a^{6} + \frac{1}{27} a^{5} - \frac{508}{12987} a^{4} + \frac{1}{9} a^{3} - \frac{1103}{25974} a^{2} - \frac{1}{6} a + \frac{3139}{8658}$, $\frac{1}{170415414} a^{19} - \frac{5}{56805138} a^{17} - \frac{14}{3155841} a^{15} - \frac{1}{13122} a^{14} - \frac{10}{1051947} a^{13} - \frac{1}{4374} a^{12} - \frac{398}{1051947} a^{11} - \frac{1}{1458} a^{10} + \frac{23}{53946} a^{9} + \frac{509}{77922} a^{7} + \frac{1}{81} a^{6} - \frac{508}{12987} a^{5} + \frac{1}{27} a^{4} - \frac{1103}{25974} a^{3} - \frac{1}{18} a^{2} + \frac{3139}{8658} a$

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:  $14$
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:  \( 3848976829.54 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:D_4$ (as 20T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 13 conjugacy class representatives for $C_5:D_4$
Character table for $C_5:D_4$

Intermediate fields

\(\Q(\sqrt{401}) \), 4.2.5788836.1, 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

Galois closure: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
401Data not computed