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Properties

Label 20.16.8193551987...8129.1
Degree $20$
Signature $[16, 2]$
Discriminant $3^{4}\cdot 97^{2}\cdot 401^{10}$
Root discriminant $39.42$
Ramified primes $3, 97, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83, -1892, 3080, 12349, -21883, -26050, 52231, 20050, -59873, 10, 37133, -8153, -12393, 4737, 2013, -1182, -93, 137, -11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 11*x^18 + 137*x^17 - 93*x^16 - 1182*x^15 + 2013*x^14 + 4737*x^13 - 12393*x^12 - 8153*x^11 + 37133*x^10 + 10*x^9 - 59873*x^8 + 20050*x^7 + 52231*x^6 - 26050*x^5 - 21883*x^4 + 12349*x^3 + 3080*x^2 - 1892*x + 83)
 
gp: K = bnfinit(x^20 - 6*x^19 - 11*x^18 + 137*x^17 - 93*x^16 - 1182*x^15 + 2013*x^14 + 4737*x^13 - 12393*x^12 - 8153*x^11 + 37133*x^10 + 10*x^9 - 59873*x^8 + 20050*x^7 + 52231*x^6 - 26050*x^5 - 21883*x^4 + 12349*x^3 + 3080*x^2 - 1892*x + 83, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 11 x^{18} + 137 x^{17} - 93 x^{16} - 1182 x^{15} + 2013 x^{14} + 4737 x^{13} - 12393 x^{12} - 8153 x^{11} + 37133 x^{10} + 10 x^{9} - 59873 x^{8} + 20050 x^{7} + 52231 x^{6} - 26050 x^{5} - 21883 x^{4} + 12349 x^{3} + 3080 x^{2} - 1892 x + 83 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(81935519873106874346065098078129=3^{4}\cdot 97^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.42$
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:  $3, 97, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{13} - \frac{1}{3} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{3} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{3} a^{9} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{16} - \frac{1}{12} a^{15} - \frac{1}{12} a^{13} - \frac{1}{6} a^{11} - \frac{1}{3} a^{10} + \frac{5}{12} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} + \frac{5}{12} a^{4} + \frac{1}{4} a^{3} + \frac{1}{3} a^{2} + \frac{1}{12} a + \frac{5}{12}$, $\frac{1}{24} a^{18} + \frac{1}{24} a^{15} + \frac{1}{24} a^{14} - \frac{1}{24} a^{13} + \frac{1}{12} a^{12} + \frac{1}{12} a^{11} + \frac{3}{8} a^{10} - \frac{3}{8} a^{9} + \frac{5}{12} a^{8} + \frac{1}{8} a^{7} - \frac{5}{24} a^{6} + \frac{1}{24} a^{5} + \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{45884936828956560819504} a^{19} - \frac{167930893994403905887}{45884936828956560819504} a^{18} - \frac{103871912356780192937}{5735617103619570102438} a^{17} + \frac{2894099912852433781841}{45884936828956560819504} a^{16} + \frac{589447872203190345617}{22942468414478280409752} a^{15} - \frac{696630036766861395401}{11471234207239140204876} a^{14} - \frac{1112331773349805274233}{15294978942985520273168} a^{13} + \frac{15812758220045576100}{955936183936595017073} a^{12} - \frac{2570262519241844120731}{15294978942985520273168} a^{11} - \frac{274305175984887962602}{955936183936595017073} a^{10} + \frac{7539272711534186815033}{45884936828956560819504} a^{9} + \frac{5481105194092345719543}{15294978942985520273168} a^{8} - \frac{9117591716226764792279}{22942468414478280409752} a^{7} + \frac{1665785398196024655223}{11471234207239140204876} a^{6} - \frac{7162354722074384958313}{45884936828956560819504} a^{5} + \frac{8973466600636881787771}{45884936828956560819504} a^{4} + \frac{4248874784833021789081}{22942468414478280409752} a^{3} + \frac{4587310713375365117975}{45884936828956560819504} a^{2} + \frac{7215172672319853113197}{45884936828956560819504} a + \frac{78717880885484222135}{184276854734765304496}$

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:  $17$
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:  \( 2029941768.34 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed