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Properties

Label 20.0.25567174371...2064.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 47^{8}$
Root discriminant $13.19$
Ramified primes $2, 47$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^4:D_5$ (as 20T73)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 4, 7, -18, 8, -12, 23, -42, 58, -54, 31, -16, 22, -20, 10, -2, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 - 2*x^17 + 10*x^16 - 20*x^15 + 22*x^14 - 16*x^13 + 31*x^12 - 54*x^11 + 58*x^10 - 42*x^9 + 23*x^8 - 12*x^7 + 8*x^6 - 18*x^5 + 7*x^4 + 4*x^3 + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 2*x^18 - 2*x^17 + 10*x^16 - 20*x^15 + 22*x^14 - 16*x^13 + 31*x^12 - 54*x^11 + 58*x^10 - 42*x^9 + 23*x^8 - 12*x^7 + 8*x^6 - 18*x^5 + 7*x^4 + 4*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 2 x^{18} - 2 x^{17} + 10 x^{16} - 20 x^{15} + 22 x^{14} - 16 x^{13} + 31 x^{12} - 54 x^{11} + 58 x^{10} - 42 x^{9} + 23 x^{8} - 12 x^{7} + 8 x^{6} - 18 x^{5} + 7 x^{4} + 4 x^{3} + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(25567174371986127192064=2^{30}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.19$
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:  $2, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{17} - \frac{2}{5} a^{16} - \frac{1}{5} a^{15} - \frac{2}{5} a^{13} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{216244080995} a^{19} - \frac{17486367282}{216244080995} a^{18} - \frac{52872851026}{216244080995} a^{17} - \frac{27177411964}{216244080995} a^{16} + \frac{38339992651}{216244080995} a^{15} + \frac{9748476028}{216244080995} a^{14} - \frac{1462325086}{9401916565} a^{13} + \frac{2036758727}{43248816199} a^{12} - \frac{7397848604}{216244080995} a^{11} - \frac{32444015774}{216244080995} a^{10} - \frac{21006897545}{43248816199} a^{9} + \frac{77399284597}{216244080995} a^{8} - \frac{80882913268}{216244080995} a^{7} + \frac{22782302981}{216244080995} a^{6} + \frac{10084938814}{43248816199} a^{5} - \frac{48785892619}{216244080995} a^{4} + \frac{58304886301}{216244080995} a^{3} + \frac{12035262784}{43248816199} a^{2} - \frac{62566786672}{216244080995} a + \frac{19049267303}{216244080995}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{4421737602}{43248816199} a^{19} + \frac{5731792916}{43248816199} a^{18} + \frac{6770252220}{43248816199} a^{17} - \frac{10404127099}{43248816199} a^{16} - \frac{24025731409}{43248816199} a^{15} + \frac{40257174570}{43248816199} a^{14} + \frac{2453925175}{1880383313} a^{13} - \frac{136197711897}{43248816199} a^{12} + \frac{65669722993}{43248816199} a^{11} + \frac{18892461292}{43248816199} a^{10} + \frac{211113703580}{43248816199} a^{9} - \frac{375264082805}{43248816199} a^{8} + \frac{431839371620}{43248816199} a^{7} - \frac{330873860156}{43248816199} a^{6} + \frac{240612203244}{43248816199} a^{5} - \frac{56120483001}{43248816199} a^{4} + \frac{82711869625}{43248816199} a^{3} - \frac{143321047843}{43248816199} a^{2} - \frac{20993594096}{43248816199} a - \frac{6965031484}{43248816199} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1468.80204297 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:D_5$ (as 20T73):

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 $C_2\times C_2^4:D_5$
Character table for $C_2\times C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.1.2209.1, 10.0.4996793344.3, 10.0.4996793344.4, 10.2.4996793344.3

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$