↑

Properties

Label 20.6.10685834074...0000.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,2^{4}\cdot 5^{14}\cdot 19^{4}\cdot 29^{6}\cdot 109^{4}$
Root discriminant $44.82$
Ramified primes $2, 5, 19, 29, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T955

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, -1303, -2259, -2820, -2620, 27251, -14078, 8098, -41312, 39740, -14259, 6616, -4695, 1579, -164, -58, -13, 5, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 + 5*x^17 - 13*x^16 - 58*x^15 - 164*x^14 + 1579*x^13 - 4695*x^12 + 6616*x^11 - 14259*x^10 + 39740*x^9 - 41312*x^8 + 8098*x^7 - 14078*x^6 + 27251*x^5 - 2620*x^4 - 2820*x^3 - 2259*x^2 - 1303*x + 31)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 + 5*x^17 - 13*x^16 - 58*x^15 - 164*x^14 + 1579*x^13 - 4695*x^12 + 6616*x^11 - 14259*x^10 + 39740*x^9 - 41312*x^8 + 8098*x^7 - 14078*x^6 + 27251*x^5 - 2620*x^4 - 2820*x^3 - 2259*x^2 - 1303*x + 31, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} + 5 x^{17} - 13 x^{16} - 58 x^{15} - 164 x^{14} + 1579 x^{13} - 4695 x^{12} + 6616 x^{11} - 14259 x^{10} + 39740 x^{9} - 41312 x^{8} + 8098 x^{7} - 14078 x^{6} + 27251 x^{5} - 2620 x^{4} - 2820 x^{3} - 2259 x^{2} - 1303 x + 31 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1068583407413817193418066406250000=-\,2^{4}\cdot 5^{14}\cdot 19^{4}\cdot 29^{6}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.82$
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:  $2, 5, 19, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{9} - \frac{1}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{14} + \frac{1}{16} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} + \frac{3}{8} a^{9} - \frac{1}{16} a^{8} + \frac{1}{4} a^{7} - \frac{7}{16} a^{6} + \frac{3}{8} a^{5} - \frac{5}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{16} a^{2} - \frac{7}{16} a - \frac{5}{16}$, $\frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{1}{16} a^{15} + \frac{3}{32} a^{14} - \frac{3}{32} a^{13} + \frac{7}{16} a^{12} - \frac{7}{16} a^{11} - \frac{1}{4} a^{10} + \frac{9}{32} a^{9} - \frac{11}{32} a^{8} + \frac{5}{32} a^{7} - \frac{3}{32} a^{6} + \frac{5}{32} a^{5} - \frac{9}{32} a^{4} - \frac{3}{32} a^{3} - \frac{3}{16} a^{2} - \frac{7}{16} a + \frac{5}{32}$, $\frac{1}{192} a^{18} + \frac{1}{96} a^{17} - \frac{5}{192} a^{16} - \frac{1}{64} a^{15} + \frac{11}{96} a^{14} + \frac{5}{192} a^{13} - \frac{7}{16} a^{12} - \frac{17}{96} a^{11} - \frac{21}{64} a^{10} - \frac{1}{3} a^{9} - \frac{1}{16} a^{8} + \frac{5}{16} a^{7} + \frac{7}{48} a^{6} + \frac{11}{96} a^{5} - \frac{13}{32} a^{4} - \frac{95}{192} a^{3} + \frac{1}{6} a^{2} - \frac{85}{192} a + \frac{31}{192}$, $\frac{1}{492436426945370607900049030630819646592} a^{19} - \frac{359631450692087378597289259693973553}{164145475648456869300016343543606548864} a^{18} + \frac{2514977933533459101793229951339438743}{164145475648456869300016343543606548864} a^{17} - \frac{2692635826512014267622862666657711795}{246218213472685303950024515315409823296} a^{16} - \frac{29038663047245896484087927533284305795}{492436426945370607900049030630819646592} a^{15} + \frac{7261215451454238725334269528898538673}{164145475648456869300016343543606548864} a^{14} + \frac{82971426451593902188167412173201847679}{492436426945370607900049030630819646592} a^{13} - \frac{114120153957800514727528268269972140539}{246218213472685303950024515315409823296} a^{12} + \frac{159356778916299682060918072801700139251}{492436426945370607900049030630819646592} a^{11} - \frac{201677967247250092328979700082903273701}{492436426945370607900049030630819646592} a^{10} + \frac{6395299215022599835198597801833962383}{61554553368171325987506128828852455824} a^{9} - \frac{338824747468650005829726400256099377}{41036368912114217325004085885901637216} a^{8} - \frac{6310420305944979311239657488609355223}{123109106736342651975012257657704911648} a^{7} - \frac{53960431705798562322305624776991005037}{246218213472685303950024515315409823296} a^{6} + \frac{3754901512969750735954650625829303711}{61554553368171325987506128828852455824} a^{5} - \frac{48557136113463802663985982285089708117}{492436426945370607900049030630819646592} a^{4} - \frac{1285182143247727474905422346196363131}{164145475648456869300016343543606548864} a^{3} + \frac{18061061983754260597357054760028217147}{492436426945370607900049030630819646592} a^{2} + \frac{8343086923616003382454087180059865495}{20518184456057108662502042942950818608} a + \frac{28207773471569169203224046828164321921}{492436426945370607900049030630819646592}$

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:  $12$
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:  \( 975258653.333 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T955:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 819200
The 275 conjugacy class representatives for t20n955 are not computed
Character table for t20n955 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.8172298511640625.1

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 $16{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R $20$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
5Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.10.0.1$x^{10} + x^{2} - 2 x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.0.1$x^{4} - x + 30$$1$$4$$0$$C_4$$[\ ]^{4}$