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Properties

Label 20.8.33966664043...5184.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 11^{16}\cdot 89^{3}$
Root discriminant $26.70$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -26, 175, -576, 730, 1534, -5650, 4438, 3080, -6248, 1860, 1196, 175, -1318, 657, 40, -104, 0, 22, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 22*x^18 - 104*x^16 + 40*x^15 + 657*x^14 - 1318*x^13 + 175*x^12 + 1196*x^11 + 1860*x^10 - 6248*x^9 + 3080*x^8 + 4438*x^7 - 5650*x^6 + 1534*x^5 + 730*x^4 - 576*x^3 + 175*x^2 - 26*x - 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 22*x^18 - 104*x^16 + 40*x^15 + 657*x^14 - 1318*x^13 + 175*x^12 + 1196*x^11 + 1860*x^10 - 6248*x^9 + 3080*x^8 + 4438*x^7 - 5650*x^6 + 1534*x^5 + 730*x^4 - 576*x^3 + 175*x^2 - 26*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 22 x^{18} - 104 x^{16} + 40 x^{15} + 657 x^{14} - 1318 x^{13} + 175 x^{12} + 1196 x^{11} + 1860 x^{10} - 6248 x^{9} + 3080 x^{8} + 4438 x^{7} - 5650 x^{6} + 1534 x^{5} + 730 x^{4} - 576 x^{3} + 175 x^{2} - 26 x - 1 \)

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:  \(33966664043402470640067805184=2^{20}\cdot 11^{16}\cdot 89^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.70$
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:  $2, 11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{4}{11} a^{14} + \frac{4}{11} a^{13} - \frac{3}{11} a^{12} - \frac{3}{11} a^{11} - \frac{4}{11} a^{10} - \frac{5}{11} a^{9} + \frac{3}{11} a^{8} - \frac{1}{11} a^{7} + \frac{1}{11} a^{6} + \frac{4}{11} a^{4} + \frac{1}{11} a^{3} + \frac{2}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} - \frac{1}{11} a^{14} + \frac{2}{11} a^{13} - \frac{4}{11} a^{12} - \frac{5}{11} a^{11} + \frac{1}{11} a^{10} + \frac{5}{11} a^{9} - \frac{3}{11} a^{7} + \frac{4}{11} a^{6} + \frac{4}{11} a^{5} - \frac{5}{11} a^{4} - \frac{5}{11} a^{3} - \frac{1}{11} a^{2} - \frac{2}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{17} - \frac{2}{11} a^{14} + \frac{3}{11} a^{12} - \frac{2}{11} a^{11} + \frac{1}{11} a^{10} - \frac{5}{11} a^{9} + \frac{3}{11} a^{7} + \frac{5}{11} a^{6} - \frac{5}{11} a^{5} - \frac{1}{11} a^{4} - \frac{5}{11} a + \frac{1}{11}$, $\frac{1}{121} a^{18} - \frac{2}{121} a^{17} + \frac{1}{121} a^{16} + \frac{2}{121} a^{15} + \frac{53}{121} a^{14} - \frac{1}{121} a^{13} - \frac{24}{121} a^{12} + \frac{32}{121} a^{11} - \frac{4}{11} a^{10} - \frac{27}{121} a^{9} - \frac{40}{121} a^{8} - \frac{30}{121} a^{7} + \frac{59}{121} a^{6} + \frac{35}{121} a^{5} + \frac{2}{121} a^{4} + \frac{21}{121} a^{3} + \frac{57}{121} a^{2} + \frac{39}{121} a + \frac{28}{121}$, $\frac{1}{12816069693765020198879} a^{19} - \frac{39987462636446985932}{12816069693765020198879} a^{18} - \frac{162115964998446264551}{12816069693765020198879} a^{17} + \frac{41635718839132197391}{12816069693765020198879} a^{16} + \frac{520118566532557641877}{12816069693765020198879} a^{15} - \frac{231532868104099829824}{1165097244887729108989} a^{14} - \frac{1714281372457345419647}{12816069693765020198879} a^{13} - \frac{198974990302039373925}{1165097244887729108989} a^{12} + \frac{5842651362626038504528}{12816069693765020198879} a^{11} + \frac{5435061516300640801781}{12816069693765020198879} a^{10} - \frac{3704648215329570865234}{12816069693765020198879} a^{9} - \frac{2655717473178996228892}{12816069693765020198879} a^{8} + \frac{2053687963973497354225}{12816069693765020198879} a^{7} - \frac{121479984999173781869}{1165097244887729108989} a^{6} - \frac{4349680778990626220745}{12816069693765020198879} a^{5} + \frac{473246222211363296914}{12816069693765020198879} a^{4} + \frac{580291435777427157396}{12816069693765020198879} a^{3} + \frac{4195224231979581267472}{12816069693765020198879} a^{2} + \frac{878191826536607697062}{12816069693765020198879} a + \frac{152002413884806355890}{12816069693765020198879}$

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:  \( 6156349.28908 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.19535810978816.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
$2$2.10.10.2$x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
2.10.10.2$x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
89Data not computed