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Properties

Label 20.8.18444893486...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{10}\cdot 11^{5}\cdot 5783^{4}$
Root discriminant $46.06$
Ramified primes $2, 5, 11, 5783$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![161051, 0, -73205, 0, -199650, 0, -211750, 0, -880, 0, 56377, 0, -80, 0, -1750, 0, -150, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 150*x^16 - 1750*x^14 - 80*x^12 + 56377*x^10 - 880*x^8 - 211750*x^6 - 199650*x^4 - 73205*x^2 + 161051)
 
gp: K = bnfinit(x^20 - 5*x^18 - 150*x^16 - 1750*x^14 - 80*x^12 + 56377*x^10 - 880*x^8 - 211750*x^6 - 199650*x^4 - 73205*x^2 + 161051, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} - 150 x^{16} - 1750 x^{14} - 80 x^{12} + 56377 x^{10} - 880 x^{8} - 211750 x^{6} - 199650 x^{4} - 73205 x^{2} + 161051 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1844489348608657807063040000000000=2^{20}\cdot 5^{10}\cdot 11^{5}\cdot 5783^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.06$
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:  $2, 5, 11, 5783$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{2} - \frac{2}{5}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{25} a^{8} - \frac{1}{25} a^{6} + \frac{1}{25} a^{4} + \frac{9}{25} a^{2} + \frac{6}{25}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{7} + \frac{1}{25} a^{5} + \frac{9}{25} a^{3} + \frac{6}{25} a$, $\frac{1}{25} a^{10} - \frac{1}{5} a^{2} - \frac{4}{25}$, $\frac{1}{25} a^{11} - \frac{1}{5} a^{3} - \frac{4}{25} a$, $\frac{1}{1375} a^{12} + \frac{6}{1375} a^{10} + \frac{3}{275} a^{8} + \frac{24}{275} a^{6} - \frac{27}{275} a^{4} + \frac{156}{1375} a^{2} - \frac{9}{125}$, $\frac{1}{1375} a^{13} + \frac{6}{1375} a^{11} + \frac{3}{275} a^{9} + \frac{24}{275} a^{7} - \frac{27}{275} a^{5} + \frac{156}{1375} a^{3} - \frac{9}{125} a$, $\frac{1}{15125} a^{14} - \frac{1}{3025} a^{12} - \frac{271}{15125} a^{10} + \frac{13}{3025} a^{8} - \frac{258}{3025} a^{6} + \frac{1201}{15125} a^{4} - \frac{1}{55} a^{2} + \frac{24}{125}$, $\frac{1}{15125} a^{15} - \frac{1}{3025} a^{13} - \frac{271}{15125} a^{11} + \frac{13}{3025} a^{9} - \frac{258}{3025} a^{7} + \frac{1201}{15125} a^{5} - \frac{1}{55} a^{3} + \frac{24}{125} a$, $\frac{1}{644703125} a^{16} - \frac{16857}{644703125} a^{14} + \frac{106253}{644703125} a^{12} - \frac{9988254}{644703125} a^{10} - \frac{1567516}{128940625} a^{8} - \frac{12058044}{644703125} a^{6} + \frac{1796533}{58609375} a^{4} + \frac{188518}{484375} a^{2} + \frac{31011}{484375}$, $\frac{1}{644703125} a^{17} - \frac{16857}{644703125} a^{15} + \frac{106253}{644703125} a^{13} - \frac{9988254}{644703125} a^{11} - \frac{1567516}{128940625} a^{9} - \frac{12058044}{644703125} a^{7} + \frac{1796533}{58609375} a^{5} + \frac{188518}{484375} a^{3} + \frac{31011}{484375} a$, $\frac{1}{1624007171875} a^{18} - \frac{36}{52387328125} a^{16} + \frac{14678591}{1624007171875} a^{14} + \frac{458938094}{1624007171875} a^{12} - \frac{12664246169}{1624007171875} a^{10} - \frac{24446091074}{1624007171875} a^{8} - \frac{5161830681}{147637015625} a^{6} - \frac{45901704}{13421546875} a^{4} + \frac{272306349}{1220140625} a^{2} + \frac{17856241}{110921875}$, $\frac{1}{1624007171875} a^{19} - \frac{36}{52387328125} a^{17} + \frac{14678591}{1624007171875} a^{15} + \frac{458938094}{1624007171875} a^{13} - \frac{12664246169}{1624007171875} a^{11} - \frac{24446091074}{1624007171875} a^{9} - \frac{5161830681}{147637015625} a^{7} - \frac{45901704}{13421546875} a^{5} + \frac{272306349}{1220140625} a^{3} + \frac{17856241}{110921875} a$

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:  $13$
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:  \( 1288238141.98 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.3.5783.1, 10.6.104509653125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$5.4.2.1$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}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5783Data not computed