↑

Properties

Label 20.20.1646157686...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{28}\cdot 5^{13}\cdot 3469^{5}$
Root discriminant $57.65$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10900, -76600, 115660, 223160, -604764, -31268, 933912, -412196, -593080, 455972, 139610, -196456, 5066, 39644, -7068, -3730, 1012, 150, -54, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 54*x^18 + 150*x^17 + 1012*x^16 - 3730*x^15 - 7068*x^14 + 39644*x^13 + 5066*x^12 - 196456*x^11 + 139610*x^10 + 455972*x^9 - 593080*x^8 - 412196*x^7 + 933912*x^6 - 31268*x^5 - 604764*x^4 + 223160*x^3 + 115660*x^2 - 76600*x + 10900)
 
gp: K = bnfinit(x^20 - 2*x^19 - 54*x^18 + 150*x^17 + 1012*x^16 - 3730*x^15 - 7068*x^14 + 39644*x^13 + 5066*x^12 - 196456*x^11 + 139610*x^10 + 455972*x^9 - 593080*x^8 - 412196*x^7 + 933912*x^6 - 31268*x^5 - 604764*x^4 + 223160*x^3 + 115660*x^2 - 76600*x + 10900, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 54 x^{18} + 150 x^{17} + 1012 x^{16} - 3730 x^{15} - 7068 x^{14} + 39644 x^{13} + 5066 x^{12} - 196456 x^{11} + 139610 x^{10} + 455972 x^{9} - 593080 x^{8} - 412196 x^{7} + 933912 x^{6} - 31268 x^{5} - 604764 x^{4} + 223160 x^{3} + 115660 x^{2} - 76600 x + 10900 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(164615768632684822200320000000000000=2^{28}\cdot 5^{13}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.65$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{10} a^{16} - \frac{1}{5} a^{15} - \frac{1}{10} a^{14} - \frac{1}{10} a^{13} - \frac{1}{10} a^{12} + \frac{1}{5} a^{11} - \frac{1}{10} a^{10} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{10} a^{17} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{2}$, $\frac{1}{10} a^{18} + \frac{1}{5} a^{15} + \frac{1}{5} a^{14} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{3}$, $\frac{1}{1086842650796669816468685503990} a^{19} + \frac{9370333950734318522695180584}{543421325398334908234342751995} a^{18} - \frac{35919790757262370437133837139}{1086842650796669816468685503990} a^{17} + \frac{4537745349957967805906409084}{543421325398334908234342751995} a^{16} + \frac{113912682312678892993267984048}{543421325398334908234342751995} a^{15} + \frac{62426693593007246260838114977}{1086842650796669816468685503990} a^{14} - \frac{93175319850920937848140624963}{543421325398334908234342751995} a^{13} - \frac{62676214441542843040580405582}{543421325398334908234342751995} a^{12} + \frac{62032061319008965882625542317}{1086842650796669816468685503990} a^{11} + \frac{173566316877635639239379094771}{1086842650796669816468685503990} a^{10} - \frac{3997996013575415959255746114}{108684265079666981646868550399} a^{9} + \frac{16830248807835263720629179858}{543421325398334908234342751995} a^{8} + \frac{52430467812701879291355812921}{543421325398334908234342751995} a^{7} - \frac{3108142941774985792778404522}{543421325398334908234342751995} a^{6} - \frac{122097679555964221976403782228}{543421325398334908234342751995} a^{5} - \frac{169421353275105520551649842246}{543421325398334908234342751995} a^{4} - \frac{237485329764457896108674691698}{543421325398334908234342751995} a^{3} + \frac{263375764697682460280050933207}{543421325398334908234342751995} a^{2} + \frac{35598071305820741557366761024}{108684265079666981646868550399} a - \frac{34000520795620346853491733432}{108684265079666981646868550399}$

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:  $19$
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:  \( 223903749476 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
3469Data not computed