↑

Properties

Label 20.14.3670901099...4375.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,5^{11}\cdot 13^{12}\cdot 19^{9}$
Root discriminant $42.49$
Ramified primes $5, 13, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T633

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 117, -284, -428, 1142, 2044, -166, -6366, -4753, 6116, 4604, -1621, -682, -587, -267, 306, -14, 16, -3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 3*x^18 + 16*x^17 - 14*x^16 + 306*x^15 - 267*x^14 - 587*x^13 - 682*x^12 - 1621*x^11 + 4604*x^10 + 6116*x^9 - 4753*x^8 - 6366*x^7 - 166*x^6 + 2044*x^5 + 1142*x^4 - 428*x^3 - 284*x^2 + 117*x - 11)
 
gp: K = bnfinit(x^20 - 5*x^19 - 3*x^18 + 16*x^17 - 14*x^16 + 306*x^15 - 267*x^14 - 587*x^13 - 682*x^12 - 1621*x^11 + 4604*x^10 + 6116*x^9 - 4753*x^8 - 6366*x^7 - 166*x^6 + 2044*x^5 + 1142*x^4 - 428*x^3 - 284*x^2 + 117*x - 11, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 3 x^{18} + 16 x^{17} - 14 x^{16} + 306 x^{15} - 267 x^{14} - 587 x^{13} - 682 x^{12} - 1621 x^{11} + 4604 x^{10} + 6116 x^{9} - 4753 x^{8} - 6366 x^{7} - 166 x^{6} + 2044 x^{5} + 1142 x^{4} - 428 x^{3} - 284 x^{2} + 117 x - 11 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-367090109903933844075668896484375=-\,5^{11}\cdot 13^{12}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.49$
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:  $5, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{116} a^{17} + \frac{1}{58} a^{16} - \frac{11}{116} a^{15} + \frac{27}{116} a^{14} + \frac{7}{58} a^{13} - \frac{9}{58} a^{12} - \frac{2}{29} a^{11} + \frac{3}{58} a^{10} - \frac{14}{29} a^{9} - \frac{11}{116} a^{8} + \frac{55}{116} a^{7} - \frac{11}{116} a^{6} + \frac{12}{29} a^{5} - \frac{7}{116} a^{4} - \frac{13}{58} a^{3} - \frac{1}{4} a^{2} - \frac{51}{116} a + \frac{23}{58}$, $\frac{1}{580} a^{18} - \frac{1}{290} a^{17} + \frac{1}{58} a^{16} + \frac{13}{580} a^{15} - \frac{13}{116} a^{14} + \frac{13}{580} a^{13} + \frac{61}{290} a^{12} - \frac{97}{290} a^{11} + \frac{47}{290} a^{10} + \frac{31}{116} a^{9} - \frac{15}{116} a^{8} + \frac{73}{290} a^{7} + \frac{1}{116} a^{6} - \frac{28}{145} a^{5} + \frac{59}{290} a^{4} + \frac{11}{29} a^{3} + \frac{239}{580} a^{2} - \frac{11}{580} a + \frac{19}{580}$, $\frac{1}{93346971448743149967020} a^{19} - \frac{7687489865784530511}{23336742862185787491755} a^{18} + \frac{74697779151484024967}{46673485724371574983510} a^{17} - \frac{2086221344114902516708}{23336742862185787491755} a^{16} - \frac{942103173570913328171}{93346971448743149967020} a^{15} + \frac{5289214998511168408797}{23336742862185787491755} a^{14} - \frac{11981887264028344092369}{93346971448743149967020} a^{13} - \frac{1212430232242241393097}{23336742862185787491755} a^{12} - \frac{23041091192314007486899}{46673485724371574983510} a^{11} - \frac{12263969633644625718923}{93346971448743149967020} a^{10} - \frac{251628471204301608033}{643772216887883792876} a^{9} + \frac{86832353540696050646}{295401808382098575845} a^{8} + \frac{13838973037175854131719}{46673485724371574983510} a^{7} + \frac{31241412753339006516793}{93346971448743149967020} a^{6} + \frac{17178572501818063814447}{93346971448743149967020} a^{5} + \frac{12392025521919366845247}{46673485724371574983510} a^{4} - \frac{6209569721494980730599}{23336742862185787491755} a^{3} + \frac{17666429475861843804151}{93346971448743149967020} a^{2} - \frac{10008798572886973777856}{23336742862185787491755} a - \frac{63941139584932374857}{3218861084439418964380}$

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:  $16$
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:  \( 2863095318.68 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T633:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40960
The 124 conjugacy class representatives for t20n633 are not computed
Character table for t20n633 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.19827925.1, 10.10.1965733049028125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R $20$ R $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
13Data not computed
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
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}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$