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Properties

Label 20.4.65861868411...0144.6
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $21.92$
Ramified primes $2, 11, 241$
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, 83, 187, -346, -1101, -351, 592, 102, 262, 200, -638, 167, 324, -307, 52, 100, -77, 7, 15, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 15*x^18 + 7*x^17 - 77*x^16 + 100*x^15 + 52*x^14 - 307*x^13 + 324*x^12 + 167*x^11 - 638*x^10 + 200*x^9 + 262*x^8 + 102*x^7 + 592*x^6 - 351*x^5 - 1101*x^4 - 346*x^3 + 187*x^2 + 83*x - 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 15*x^18 + 7*x^17 - 77*x^16 + 100*x^15 + 52*x^14 - 307*x^13 + 324*x^12 + 167*x^11 - 638*x^10 + 200*x^9 + 262*x^8 + 102*x^7 + 592*x^6 - 351*x^5 - 1101*x^4 - 346*x^3 + 187*x^2 + 83*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 15 x^{18} + 7 x^{17} - 77 x^{16} + 100 x^{15} + 52 x^{14} - 307 x^{13} + 324 x^{12} + 167 x^{11} - 638 x^{10} + 200 x^{9} + 262 x^{8} + 102 x^{7} + 592 x^{6} - 351 x^{5} - 1101 x^{4} - 346 x^{3} + 187 x^{2} + 83 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(658618684118710741619590144=2^{10}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.92$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{14} + \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{3}{8} a^{10} - \frac{1}{2} a^{9} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{8} a^{2} + \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{13} + \frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{3}{8} a^{10} - \frac{1}{2} a^{8} - \frac{3}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{17} - \frac{1}{16} a^{14} + \frac{1}{8} a^{13} - \frac{3}{16} a^{12} + \frac{5}{16} a^{11} - \frac{3}{16} a^{10} + \frac{1}{4} a^{9} + \frac{1}{16} a^{8} - \frac{3}{8} a^{7} - \frac{7}{16} a^{6} + \frac{1}{8} a^{5} + \frac{7}{16} a^{4} + \frac{1}{16} a^{3} - \frac{3}{8} a^{2} - \frac{7}{16} a - \frac{5}{16}$, $\frac{1}{7624292183605613104} a^{19} + \frac{43871747780442639}{1906073045901403276} a^{18} + \frac{322043306587557545}{7624292183605613104} a^{17} - \frac{9080967348053750}{476518261475350819} a^{16} - \frac{223267256050593157}{7624292183605613104} a^{15} + \frac{658985029868448941}{7624292183605613104} a^{14} + \frac{1660184845553697845}{7624292183605613104} a^{13} - \frac{228156372527312183}{1906073045901403276} a^{12} + \frac{1561094895356751753}{3812146091802806552} a^{11} + \frac{1204020636450019259}{7624292183605613104} a^{10} - \frac{2088943308355256715}{7624292183605613104} a^{9} - \frac{89922862156495593}{7624292183605613104} a^{8} - \frac{5566622295797297}{17367408163110736} a^{7} + \frac{520854899595970025}{7624292183605613104} a^{6} - \frac{989333302653690435}{7624292183605613104} a^{5} + \frac{901592247251076131}{1906073045901403276} a^{4} + \frac{2790117964231238489}{7624292183605613104} a^{3} + \frac{2261243989107171171}{7624292183605613104} a^{2} - \frac{122397617917474477}{476518261475350819} a - \frac{2686635509483961255}{7624292183605613104}$

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:  \( 208707.57778 \) (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

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11Data not computed
241Data not computed