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Properties

Label 20.4.15418170861...0000.4
Degree $20$
Signature $[4, 8]$
Discriminant $2^{30}\cdot 5^{5}\cdot 11^{16}$
Root discriminant $28.80$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 22, -110, 250, -219, 660, -2098, 328, 5119, -3230, -2358, 2480, -81, -732, 454, -130, -16, 16, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 2*x^18 + 16*x^17 - 16*x^16 - 130*x^15 + 454*x^14 - 732*x^13 - 81*x^12 + 2480*x^11 - 2358*x^10 - 3230*x^9 + 5119*x^8 + 328*x^7 - 2098*x^6 + 660*x^5 - 219*x^4 + 250*x^3 - 110*x^2 + 22*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 2*x^18 + 16*x^17 - 16*x^16 - 130*x^15 + 454*x^14 - 732*x^13 - 81*x^12 + 2480*x^11 - 2358*x^10 - 3230*x^9 + 5119*x^8 + 328*x^7 - 2098*x^6 + 660*x^5 - 219*x^4 + 250*x^3 - 110*x^2 + 22*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 2 x^{18} + 16 x^{17} - 16 x^{16} - 130 x^{15} + 454 x^{14} - 732 x^{13} - 81 x^{12} + 2480 x^{11} - 2358 x^{10} - 3230 x^{9} + 5119 x^{8} + 328 x^{7} - 2098 x^{6} + 660 x^{5} - 219 x^{4} + 250 x^{3} - 110 x^{2} + 22 x - 1 \)

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:  \(154181708612560135336755200000=2^{30}\cdot 5^{5}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.80$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{20742847582881296750187414788218919} a^{19} - \frac{3863818690718551053309953162843592}{20742847582881296750187414788218919} a^{18} + \frac{6346212856484119864911055696135493}{20742847582881296750187414788218919} a^{17} + \frac{1819909617930388757438121293081984}{20742847582881296750187414788218919} a^{16} - \frac{3622785184999354916538410224375429}{20742847582881296750187414788218919} a^{15} + \frac{6821078063549636212172174163804185}{20742847582881296750187414788218919} a^{14} - \frac{2831603881199325966715082131243093}{20742847582881296750187414788218919} a^{13} + \frac{2115845545598792564811563310946988}{20742847582881296750187414788218919} a^{12} - \frac{2120377855118631117951857910250810}{20742847582881296750187414788218919} a^{11} - \frac{5634687102228017035432915099707533}{20742847582881296750187414788218919} a^{10} - \frac{140569951067754548147790611283068}{309594740043004429107274847585357} a^{9} + \frac{3412317178659978721086714195257665}{20742847582881296750187414788218919} a^{8} + \frac{5379019729041770684582404348916528}{20742847582881296750187414788218919} a^{7} - \frac{9993705897653691945165441651543595}{20742847582881296750187414788218919} a^{6} - \frac{9708671819400035541761990198746569}{20742847582881296750187414788218919} a^{5} - \frac{1660343505049736414626907600941274}{20742847582881296750187414788218919} a^{4} - \frac{5872168989175352770415133446914022}{20742847582881296750187414788218919} a^{3} - \frac{6878834324665857409301056991658525}{20742847582881296750187414788218919} a^{2} + \frac{7120968574474181508762629684638399}{20742847582881296750187414788218919} a + \frac{3060472662406385998417937630157689}{20742847582881296750187414788218919}$

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:  \( 4337847.57508 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T427:

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 t20n427 are not computed
Character table for t20n427 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.219503494144.2

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 $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.0.1$x^{10} + x^{2} - x + 3$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$