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Properties

Label 20.0.41026450600...6416.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{32}\cdot 13^{12}\cdot 41$
Root discriminant $17.01$
Ramified primes $2, 13, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 48, 104, 96, 48, 8, -64, -16, 24, -88, 64, -76, 76, -16, 16, -4, -4, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^17 - 4*x^16 - 4*x^15 + 16*x^14 - 16*x^13 + 76*x^12 - 76*x^11 + 64*x^10 - 88*x^9 + 24*x^8 - 16*x^7 - 64*x^6 + 8*x^5 + 48*x^4 + 96*x^3 + 104*x^2 + 48*x + 8)
 
gp: K = bnfinit(x^20 - 4*x^17 - 4*x^16 - 4*x^15 + 16*x^14 - 16*x^13 + 76*x^12 - 76*x^11 + 64*x^10 - 88*x^9 + 24*x^8 - 16*x^7 - 64*x^6 + 8*x^5 + 48*x^4 + 96*x^3 + 104*x^2 + 48*x + 8, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{17} - 4 x^{16} - 4 x^{15} + 16 x^{14} - 16 x^{13} + 76 x^{12} - 76 x^{11} + 64 x^{10} - 88 x^{9} + 24 x^{8} - 16 x^{7} - 64 x^{6} + 8 x^{5} + 48 x^{4} + 96 x^{3} + 104 x^{2} + 48 x + 8 \)

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:  \(4102645060079682024636416=2^{32}\cdot 13^{12}\cdot 41\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.01$
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:  $2, 13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{4} a^{14}$, $\frac{1}{4} a^{15}$, $\frac{1}{4} a^{16}$, $\frac{1}{4} a^{17}$, $\frac{1}{4} a^{18}$, $\frac{1}{734089615572620} a^{19} - \frac{12773207374759}{183522403893155} a^{18} - \frac{18503400051996}{183522403893155} a^{17} + \frac{12413173104749}{146817923114524} a^{16} - \frac{37121020093777}{367044807786310} a^{15} - \frac{2265493060615}{36704480778631} a^{14} - \frac{23681861445247}{367044807786310} a^{13} + \frac{5390985377537}{183522403893155} a^{12} - \frac{83458778502201}{367044807786310} a^{11} + \frac{85058099090373}{367044807786310} a^{10} + \frac{34603546160967}{183522403893155} a^{9} - \frac{2189308748183}{367044807786310} a^{8} - \frac{8593532048030}{36704480778631} a^{7} + \frac{56908965828586}{183522403893155} a^{6} - \frac{34625920987787}{183522403893155} a^{5} - \frac{53155767605016}{183522403893155} a^{4} - \frac{15127499250622}{183522403893155} a^{3} + \frac{49664696622451}{183522403893155} a^{2} + \frac{6965666902553}{36704480778631} a + \frac{35180230287667}{183522403893155}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{3822567689}{2264799974} a^{19} + \frac{994198175}{1132399987} a^{18} - \frac{951095105}{2264799974} a^{17} + \frac{15696721165}{2264799974} a^{16} + \frac{7195777117}{2264799974} a^{15} + \frac{5589487209}{1132399987} a^{14} - \frac{66941813517}{2264799974} a^{13} + \frac{95702696129}{2264799974} a^{12} - \frac{338761903527}{2264799974} a^{11} + \frac{463756804599}{2264799974} a^{10} - \frac{238477842020}{1132399987} a^{9} + \frac{284820419296}{1132399987} a^{8} - \frac{184911494283}{1132399987} a^{7} + \frac{116158348576}{1132399987} a^{6} + \frac{71376854775}{1132399987} a^{5} - \frac{59279060568}{1132399987} a^{4} - \frac{59411647788}{1132399987} a^{3} - \frac{151714248036}{1132399987} a^{2} - \frac{120100557268}{1132399987} a - \frac{26900452559}{1132399987} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 55854.4013726 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.1.35152.1, 10.0.79082438656.1

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 $20$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.0.1$x^{8} - x + 12$$1$$8$$0$$C_8$$[\ ]^{8}$