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Properties

Label 20.12.8169342613...7408.2
Degree $20$
Signature $[12, 4]$
Discriminant $2^{10}\cdot 3^{15}\cdot 11^{18}$
Root discriminant $27.90$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T409

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 14, 52, 76, 198, 430, -547, -1620, 923, 1804, -912, -704, -12, 470, 36, -186, 44, -12, 19, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 19*x^18 - 12*x^17 + 44*x^16 - 186*x^15 + 36*x^14 + 470*x^13 - 12*x^12 - 704*x^11 - 912*x^10 + 1804*x^9 + 923*x^8 - 1620*x^7 - 547*x^6 + 430*x^5 + 198*x^4 + 76*x^3 + 52*x^2 + 14*x + 1)
 
gp: K = bnfinit(x^20 - 8*x^19 + 19*x^18 - 12*x^17 + 44*x^16 - 186*x^15 + 36*x^14 + 470*x^13 - 12*x^12 - 704*x^11 - 912*x^10 + 1804*x^9 + 923*x^8 - 1620*x^7 - 547*x^6 + 430*x^5 + 198*x^4 + 76*x^3 + 52*x^2 + 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 19 x^{18} - 12 x^{17} + 44 x^{16} - 186 x^{15} + 36 x^{14} + 470 x^{13} - 12 x^{12} - 704 x^{11} - 912 x^{10} + 1804 x^{9} + 923 x^{8} - 1620 x^{7} - 547 x^{6} + 430 x^{5} + 198 x^{4} + 76 x^{3} + 52 x^{2} + 14 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(81693426134005631737181457408=2^{10}\cdot 3^{15}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.90$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{268} a^{17} + \frac{13}{134} a^{16} + \frac{3}{67} a^{15} + \frac{55}{268} a^{14} - \frac{37}{268} a^{13} - \frac{32}{67} a^{12} - \frac{24}{67} a^{11} - \frac{45}{134} a^{10} - \frac{39}{134} a^{9} + \frac{87}{268} a^{8} - \frac{21}{134} a^{7} - \frac{117}{268} a^{6} - \frac{21}{134} a^{5} - \frac{13}{67} a^{4} - \frac{107}{268} a^{3} + \frac{29}{268} a^{2} - \frac{22}{67} a + \frac{16}{67}$, $\frac{1}{268} a^{18} + \frac{3}{134} a^{16} + \frac{11}{268} a^{15} - \frac{127}{268} a^{14} - \frac{26}{67} a^{13} - \frac{59}{134} a^{12} - \frac{3}{134} a^{11} + \frac{59}{134} a^{10} - \frac{29}{268} a^{9} + \frac{27}{67} a^{8} + \frac{37}{268} a^{7} + \frac{13}{67} a^{6} + \frac{51}{134} a^{5} - \frac{95}{268} a^{4} + \frac{131}{268} a^{3} + \frac{24}{67} a^{2} + \frac{37}{134} a - \frac{14}{67}$, $\frac{1}{64514615520956} a^{19} - \frac{53118774617}{32257307760478} a^{18} + \frac{14566201742}{16128653880239} a^{17} - \frac{3043825493777}{32257307760478} a^{16} - \frac{3081506613539}{32257307760478} a^{15} - \frac{179145853365}{481452354634} a^{14} - \frac{23645026642427}{64514615520956} a^{13} - \frac{7254692517471}{32257307760478} a^{12} + \frac{28964265058333}{64514615520956} a^{11} - \frac{223553408807}{962904709268} a^{10} - \frac{2182745531695}{16128653880239} a^{9} - \frac{17988660305097}{64514615520956} a^{8} + \frac{27422603110733}{64514615520956} a^{7} + \frac{30647992178565}{64514615520956} a^{6} - \frac{1339711309655}{16128653880239} a^{5} - \frac{69505666829}{481452354634} a^{4} - \frac{16095565827169}{32257307760478} a^{3} + \frac{16511375339447}{64514615520956} a^{2} - \frac{1295558874236}{16128653880239} a - \frac{11731282578855}{64514615520956}$

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:  $15$
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:  \( 17434417.9908 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T409:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n409 are not computed
Character table for t20n409 is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\)

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 R $20$ $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.12$x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
3Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$