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Properties

Label 20.4.20245771322...4048.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{30}\cdot 3^{18}\cdot 13^{5}\cdot 107^{4}$
Root discriminant $36.76$
Ramified primes $2, 3, 13, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1023

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-183, 876, -5310, 4644, 3280, -7270, 6732, -4512, -849, 7344, -10414, 9562, -6993, 4120, -1720, 324, 129, -126, 48, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 48*x^18 - 126*x^17 + 129*x^16 + 324*x^15 - 1720*x^14 + 4120*x^13 - 6993*x^12 + 9562*x^11 - 10414*x^10 + 7344*x^9 - 849*x^8 - 4512*x^7 + 6732*x^6 - 7270*x^5 + 3280*x^4 + 4644*x^3 - 5310*x^2 + 876*x - 183)
 
gp: K = bnfinit(x^20 - 10*x^19 + 48*x^18 - 126*x^17 + 129*x^16 + 324*x^15 - 1720*x^14 + 4120*x^13 - 6993*x^12 + 9562*x^11 - 10414*x^10 + 7344*x^9 - 849*x^8 - 4512*x^7 + 6732*x^6 - 7270*x^5 + 3280*x^4 + 4644*x^3 - 5310*x^2 + 876*x - 183, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 48 x^{18} - 126 x^{17} + 129 x^{16} + 324 x^{15} - 1720 x^{14} + 4120 x^{13} - 6993 x^{12} + 9562 x^{11} - 10414 x^{10} + 7344 x^{9} - 849 x^{8} - 4512 x^{7} + 6732 x^{6} - 7270 x^{5} + 3280 x^{4} + 4644 x^{3} - 5310 x^{2} + 876 x - 183 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(20245771322351194925844343554048=2^{30}\cdot 3^{18}\cdot 13^{5}\cdot 107^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.76$
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:  $2, 3, 13, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{10} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{11} + \frac{1}{3} a^{5}$, $\frac{1}{39} a^{18} + \frac{5}{39} a^{16} + \frac{2}{39} a^{15} - \frac{1}{39} a^{14} - \frac{19}{39} a^{13} + \frac{6}{13} a^{12} + \frac{8}{39} a^{11} - \frac{17}{39} a^{10} + \frac{1}{3} a^{9} - \frac{11}{39} a^{8} + \frac{19}{39} a^{7} - \frac{4}{39} a^{6} + \frac{1}{3} a^{5} - \frac{4}{13} a^{4} - \frac{2}{13} a^{3} + \frac{6}{13} a^{2} + \frac{4}{13} a + \frac{1}{13}$, $\frac{1}{925769085757417087988603866857} a^{19} - \frac{89305062190420373317936298}{21529513622265513674153578299} a^{18} - \frac{113132259951714976292924398387}{925769085757417087988603866857} a^{17} - \frac{15737924656586593090371122252}{925769085757417087988603866857} a^{16} - \frac{72455907552627793516755401797}{925769085757417087988603866857} a^{15} - \frac{94810811208192934189378944272}{925769085757417087988603866857} a^{14} + \frac{374679157431632217184573491880}{925769085757417087988603866857} a^{13} + \frac{134616802368740794500689629409}{308589695252472362662867955619} a^{12} - \frac{34192096466682891437781682553}{102863231750824120887622651873} a^{11} + \frac{319949756152070361620970981358}{925769085757417087988603866857} a^{10} - \frac{159257144668397067768490937414}{925769085757417087988603866857} a^{9} - \frac{180784166094928713000315739513}{925769085757417087988603866857} a^{8} + \frac{305252423096159882427249486823}{925769085757417087988603866857} a^{7} + \frac{44759385331209580351223631278}{925769085757417087988603866857} a^{6} - \frac{251455447852714313672734797617}{925769085757417087988603866857} a^{5} - \frac{255604299019098148544872735175}{925769085757417087988603866857} a^{4} - \frac{124040510364972149744286848836}{308589695252472362662867955619} a^{3} + \frac{58616125979190490754743264867}{308589695252472362662867955619} a^{2} - \frac{103805637071022168355429084213}{308589695252472362662867955619} a - \frac{55217813139954273751558487701}{308589695252472362662867955619}$

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:  $11$
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:  \( 131920417.361 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1023:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 324 conjugacy class representatives for t20n1023 are not computed
Character table for t20n1023 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.6.38998285028352.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.12.12.28$x^{12} + 12 x^{11} - 3 x^{10} + 3 x^{9} + 3 x^{8} + 6 x^{7} + 12 x^{6} + 9 x^{5} + 9 x^{4} + 9 x + 9$$6$$2$$12$12T34$[5/4, 5/4]_{4}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\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$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
107Data not computed