↑

Properties

Label 20.4.32473988377...0464.4
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{17}\cdot 89^{4}$
Root discriminant $26.64$
Ramified primes $2, 11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T427

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-617, -339, 1163, 153, -528, 315, -1010, -586, 1051, -253, -307, 618, -353, -348, 376, 12, -127, 43, 4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 4*x^18 + 43*x^17 - 127*x^16 + 12*x^15 + 376*x^14 - 348*x^13 - 353*x^12 + 618*x^11 - 307*x^10 - 253*x^9 + 1051*x^8 - 586*x^7 - 1010*x^6 + 315*x^5 - 528*x^4 + 153*x^3 + 1163*x^2 - 339*x - 617)
 
gp: K = bnfinit(x^20 - 5*x^19 + 4*x^18 + 43*x^17 - 127*x^16 + 12*x^15 + 376*x^14 - 348*x^13 - 353*x^12 + 618*x^11 - 307*x^10 - 253*x^9 + 1051*x^8 - 586*x^7 - 1010*x^6 + 315*x^5 - 528*x^4 + 153*x^3 + 1163*x^2 - 339*x - 617, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 4 x^{18} + 43 x^{17} - 127 x^{16} + 12 x^{15} + 376 x^{14} - 348 x^{13} - 353 x^{12} + 618 x^{11} - 307 x^{10} - 253 x^{9} + 1051 x^{8} - 586 x^{7} - 1010 x^{6} + 315 x^{5} - 528 x^{4} + 153 x^{3} + 1163 x^{2} - 339 x - 617 \)

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:  \(32473988377432635504517950464=2^{10}\cdot 11^{17}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.64$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{9031418756746729997013006235677461} a^{19} + \frac{1830061576735502761719483482037567}{9031418756746729997013006235677461} a^{18} + \frac{3740611108177266559851303533779754}{9031418756746729997013006235677461} a^{17} + \frac{3265270787449439089152884376448787}{9031418756746729997013006235677461} a^{16} - \frac{1359518770032124707050923769650434}{9031418756746729997013006235677461} a^{15} - \frac{334929006619251795519314223149118}{9031418756746729997013006235677461} a^{14} - \frac{362891646592179126074412963083921}{9031418756746729997013006235677461} a^{13} - \frac{1863717764342950235871485568875446}{9031418756746729997013006235677461} a^{12} + \frac{3419365690041221420546900766929573}{9031418756746729997013006235677461} a^{11} + \frac{2690899494650376666423137641670885}{9031418756746729997013006235677461} a^{10} - \frac{1589943982321963934079271379401589}{9031418756746729997013006235677461} a^{9} + \frac{4272538388290090436132676588056551}{9031418756746729997013006235677461} a^{8} + \frac{1626531328389953052014556586409701}{9031418756746729997013006235677461} a^{7} - \frac{3640951052226090670799582533013649}{9031418756746729997013006235677461} a^{6} - \frac{2858824363804584931336364797390734}{9031418756746729997013006235677461} a^{5} + \frac{2590767079672940313571109372868606}{9031418756746729997013006235677461} a^{4} + \frac{72664934372986351105795131950695}{9031418756746729997013006235677461} a^{3} - \frac{4325520339266530666648235914070389}{9031418756746729997013006235677461} a^{2} + \frac{2285526271887663105936237580820822}{9031418756746729997013006235677461} a + \frac{522346568865412710715784282360983}{9031418756746729997013006235677461}$

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:  \( 2111445.9982 \) (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

\(\Q(\zeta_{11})^+\), 10.2.19077940409.1

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
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 $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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$2.10.10.7$x^{10} - x^{8} - x^{6} - 3 x^{2} - 7$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$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}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
89Data not computed