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Properties

Label 20.0.84141446660...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{15}\cdot 29^{10}$
Root discriminant $31.35$
Ramified primes $2, 5, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18671, 2366, -10375, -18754, -11105, 22302, 25816, -34154, 13461, -8834, 6539, -2716, 1851, -1176, 466, -242, 115, -36, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 15*x^18 - 36*x^17 + 115*x^16 - 242*x^15 + 466*x^14 - 1176*x^13 + 1851*x^12 - 2716*x^11 + 6539*x^10 - 8834*x^9 + 13461*x^8 - 34154*x^7 + 25816*x^6 + 22302*x^5 - 11105*x^4 - 18754*x^3 - 10375*x^2 + 2366*x + 18671)
 
gp: K = bnfinit(x^20 - 6*x^19 + 15*x^18 - 36*x^17 + 115*x^16 - 242*x^15 + 466*x^14 - 1176*x^13 + 1851*x^12 - 2716*x^11 + 6539*x^10 - 8834*x^9 + 13461*x^8 - 34154*x^7 + 25816*x^6 + 22302*x^5 - 11105*x^4 - 18754*x^3 - 10375*x^2 + 2366*x + 18671, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 15 x^{18} - 36 x^{17} + 115 x^{16} - 242 x^{15} + 466 x^{14} - 1176 x^{13} + 1851 x^{12} - 2716 x^{11} + 6539 x^{10} - 8834 x^{9} + 13461 x^{8} - 34154 x^{7} + 25816 x^{6} + 22302 x^{5} - 11105 x^{4} - 18754 x^{3} - 10375 x^{2} + 2366 x + 18671 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(841414466600402000000000000000=2^{16}\cdot 5^{15}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.35$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} - \frac{1}{2} a^{9} + \frac{1}{10} a^{8} - \frac{2}{5} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{11} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{10} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{10} a^{14} + \frac{1}{5} a^{11} + \frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{1}{2} a^{8} + \frac{1}{5} a^{7} - \frac{1}{2} a^{6} + \frac{1}{5} a^{5} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{10}$, $\frac{1}{50} a^{15} - \frac{1}{50} a^{14} - \frac{1}{50} a^{12} - \frac{1}{5} a^{11} + \frac{3}{25} a^{10} + \frac{3}{50} a^{9} + \frac{12}{25} a^{8} - \frac{1}{25} a^{7} - \frac{8}{25} a^{6} - \frac{23}{50} a^{5} + \frac{1}{5} a^{4} + \frac{13}{50} a^{3} - \frac{1}{10} a^{2} - \frac{17}{50} a + \frac{3}{50}$, $\frac{1}{50} a^{16} - \frac{1}{50} a^{14} - \frac{1}{50} a^{13} - \frac{1}{50} a^{12} + \frac{1}{50} a^{11} + \frac{9}{50} a^{10} - \frac{23}{50} a^{9} + \frac{7}{50} a^{8} + \frac{7}{50} a^{7} + \frac{21}{50} a^{6} - \frac{13}{50} a^{5} + \frac{4}{25} a^{4} - \frac{17}{50} a^{3} - \frac{11}{25} a^{2} + \frac{3}{25} a - \frac{6}{25}$, $\frac{1}{50} a^{17} - \frac{1}{25} a^{14} - \frac{1}{50} a^{13} - \frac{1}{50} a^{11} + \frac{4}{25} a^{10} - \frac{3}{10} a^{9} + \frac{3}{25} a^{8} + \frac{19}{50} a^{7} - \frac{2}{25} a^{6} + \frac{1}{5} a^{5} + \frac{9}{25} a^{4} - \frac{9}{50} a^{3} - \frac{12}{25} a^{2} - \frac{2}{25} a - \frac{11}{25}$, $\frac{1}{159500} a^{18} + \frac{179}{31900} a^{17} - \frac{139}{159500} a^{16} - \frac{31}{15950} a^{15} + \frac{619}{39875} a^{14} - \frac{1094}{39875} a^{13} + \frac{883}{79750} a^{12} - \frac{4204}{39875} a^{11} + \frac{7691}{159500} a^{10} - \frac{6091}{14500} a^{9} - \frac{60421}{159500} a^{8} - \frac{12109}{39875} a^{7} - \frac{24603}{79750} a^{6} - \frac{17074}{39875} a^{5} + \frac{2797}{79750} a^{4} - \frac{473}{1450} a^{3} + \frac{59099}{159500} a^{2} - \frac{857}{31900} a + \frac{37289}{159500}$, $\frac{1}{33677152551206397244300266059500} a^{19} + \frac{12566009199512244838560623}{8419288137801599311075066514875} a^{18} + \frac{1719990307253875822570517929}{8419288137801599311075066514875} a^{17} - \frac{71834909305025614325688398073}{33677152551206397244300266059500} a^{16} - \frac{54479991485901684533372199137}{16838576275603198622150133029750} a^{15} + \frac{20220924841074961488080006647}{580640561227696504212073552750} a^{14} - \frac{413879867168746994703278901833}{16838576275603198622150133029750} a^{13} + \frac{211421711924516425955426427923}{16838576275603198622150133029750} a^{12} - \frac{3469170294480437166390368615031}{33677152551206397244300266059500} a^{11} + \frac{2922998795218157096081682077423}{16838576275603198622150133029750} a^{10} - \frac{1802394517976951019614030907242}{8419288137801599311075066514875} a^{9} - \frac{2196534806365240085315235063563}{33677152551206397244300266059500} a^{8} - \frac{4207368528055705084619278683022}{8419288137801599311075066514875} a^{7} - \frac{3401449046756605956366230379719}{16838576275603198622150133029750} a^{6} - \frac{1926719085263629582639744051172}{8419288137801599311075066514875} a^{5} + \frac{1636923024286526129813821911459}{16838576275603198622150133029750} a^{4} + \frac{8577658973991559480401974237699}{33677152551206397244300266059500} a^{3} + \frac{1420066704411022301029391560567}{8419288137801599311075066514875} a^{2} + \frac{1099491108222196663306808108457}{16838576275603198622150133029750} a - \frac{12021666370641981797331579976877}{33677152551206397244300266059500}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 2498106.65213 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.105125.2, 5.1.1682000.2 x5, 10.2.14145620000000.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: 5.1.1682000.2
Degree 10 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$