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Properties

Label 20.0.87960930222...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{56}\cdot 5^{13}$
Root discriminant $19.83$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 32, 96, 92, -110, -256, -80, 96, 116, 16, -20, 8, -16, 16, -4, 4, 12, -8, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 4*x^18 - 8*x^17 + 12*x^16 + 4*x^15 - 4*x^14 + 16*x^13 - 16*x^12 + 8*x^11 - 20*x^10 + 16*x^9 + 116*x^8 + 96*x^7 - 80*x^6 - 256*x^5 - 110*x^4 + 92*x^3 + 96*x^2 + 32*x + 4)
 
gp: K = bnfinit(x^20 - 2*x^19 + 4*x^18 - 8*x^17 + 12*x^16 + 4*x^15 - 4*x^14 + 16*x^13 - 16*x^12 + 8*x^11 - 20*x^10 + 16*x^9 + 116*x^8 + 96*x^7 - 80*x^6 - 256*x^5 - 110*x^4 + 92*x^3 + 96*x^2 + 32*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 4 x^{18} - 8 x^{17} + 12 x^{16} + 4 x^{15} - 4 x^{14} + 16 x^{13} - 16 x^{12} + 8 x^{11} - 20 x^{10} + 16 x^{9} + 116 x^{8} + 96 x^{7} - 80 x^{6} - 256 x^{5} - 110 x^{4} + 92 x^{3} + 96 x^{2} + 32 x + 4 \)

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:  \(87960930222080000000000000=2^{56}\cdot 5^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.83$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{10} a^{18} + \frac{1}{5} a^{17} - \frac{1}{5} a^{16} - \frac{2}{5} a^{15} + \frac{2}{5} a^{14} - \frac{2}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{249707776330894730} a^{19} + \frac{5274905533165076}{124853888165447365} a^{18} + \frac{30226336088178923}{249707776330894730} a^{17} - \frac{12608280908847372}{124853888165447365} a^{16} + \frac{43532253808093717}{124853888165447365} a^{15} + \frac{12567317676102218}{124853888165447365} a^{14} + \frac{56444277708321412}{124853888165447365} a^{13} + \frac{19466009430558139}{124853888165447365} a^{12} - \frac{9407942107378750}{24970777633089473} a^{11} + \frac{18497499811307523}{124853888165447365} a^{10} + \frac{42134012097344317}{124853888165447365} a^{9} + \frac{22121555596856874}{124853888165447365} a^{8} + \frac{35418273818937731}{124853888165447365} a^{7} + \frac{6542332683378206}{124853888165447365} a^{6} - \frac{4374235616381434}{24970777633089473} a^{5} - \frac{4283304164770102}{124853888165447365} a^{4} + \frac{52981921379547152}{124853888165447365} a^{3} + \frac{2855449698129282}{124853888165447365} a^{2} - \frac{45238836612433392}{124853888165447365} a + \frac{1434874512343654}{24970777633089473}$

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:  \( \frac{45258167051}{16663309618} a^{19} - \frac{113239975663}{16663309618} a^{18} + \frac{119187867018}{8331654809} a^{17} - \frac{240867577786}{8331654809} a^{16} + \frac{392779768367}{8331654809} a^{15} - \frac{106952084025}{8331654809} a^{14} - \frac{36847345918}{8331654809} a^{13} + \frac{385760509077}{8331654809} a^{12} - \frac{552153782702}{8331654809} a^{11} + \frac{462593306780}{8331654809} a^{10} - \frac{676506324206}{8331654809} a^{9} + \frac{701851847267}{8331654809} a^{8} + \frac{2275668406982}{8331654809} a^{7} + \frac{1028481724792}{8331654809} a^{6} - \frac{2292679230388}{8331654809} a^{5} - \frac{4556707111782}{8331654809} a^{4} - \frac{137059949045}{8331654809} a^{3} + \frac{2120625625589}{8331654809} a^{2} + \frac{1008938119192}{8331654809} a + \frac{160301167397}{8331654809} \) (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:  \( 547846.080547 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.320.1, 5.1.256000.1, 10.0.1048576000000.1

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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$