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Properties

Label 20.0.81976336236...8176.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 163^{10}$
Root discriminant $44.23$
Ramified primes $2, 3, 163$
Class number $242$ (GRH)
Class group $[11, 22]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![352836, 0, 835704, 0, 977697, 0, 1060704, 0, 603766, 0, 160128, 0, 14355, 0, -1208, 0, -234, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 234*x^16 - 1208*x^14 + 14355*x^12 + 160128*x^10 + 603766*x^8 + 1060704*x^6 + 977697*x^4 + 835704*x^2 + 352836)
 
gp: K = bnfinit(x^20 - 234*x^16 - 1208*x^14 + 14355*x^12 + 160128*x^10 + 603766*x^8 + 1060704*x^6 + 977697*x^4 + 835704*x^2 + 352836, 1)
 

Normalized defining polynomial

\( x^{20} - 234 x^{16} - 1208 x^{14} + 14355 x^{12} + 160128 x^{10} + 603766 x^{8} + 1060704 x^{6} + 977697 x^{4} + 835704 x^{2} + 352836 \)

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:  \(819763362363033867895669628338176=2^{20}\cdot 3^{10}\cdot 163^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.23$
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:  $2, 3, 163$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3}$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{5}{12} a^{5} - \frac{1}{2} a^{4} - \frac{1}{12} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2}$, $\frac{1}{36} a^{9} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{5}{12} a^{5} - \frac{1}{12} a^{4} - \frac{4}{9} a^{3} + \frac{5}{12} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{36} a^{10} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{7}{18} a^{4} - \frac{1}{2} a^{3} + \frac{5}{12} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{72} a^{11} + \frac{1}{24} a^{7} - \frac{1}{12} a^{6} + \frac{5}{18} a^{5} + \frac{5}{12} a^{4} + \frac{7}{24} a^{3} - \frac{1}{12} a^{2} - \frac{1}{4} a$, $\frac{1}{72} a^{12} - \frac{1}{24} a^{8} - \frac{1}{18} a^{6} + \frac{11}{24} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{72} a^{13} - \frac{1}{72} a^{9} + \frac{1}{36} a^{7} - \frac{1}{12} a^{6} + \frac{5}{24} a^{5} - \frac{1}{12} a^{4} + \frac{2}{9} a^{3} + \frac{5}{12} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{432} a^{14} - \frac{1}{144} a^{13} + \frac{1}{432} a^{12} + \frac{1}{432} a^{10} - \frac{1}{144} a^{9} - \frac{7}{432} a^{8} - \frac{1}{18} a^{7} - \frac{19}{432} a^{6} + \frac{1}{48} a^{5} + \frac{89}{432} a^{4} + \frac{1}{36} a^{3} + \frac{2}{9} a^{2} + \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{4752} a^{15} + \frac{1}{1188} a^{13} - \frac{1}{144} a^{12} + \frac{31}{4752} a^{11} + \frac{1}{108} a^{9} + \frac{1}{48} a^{8} - \frac{157}{4752} a^{7} + \frac{1}{36} a^{6} + \frac{31}{108} a^{5} + \frac{13}{48} a^{4} - \frac{43}{88} a^{3} + \frac{1}{4} a^{2} - \frac{65}{132} a - \frac{1}{4}$, $\frac{1}{38016} a^{16} + \frac{37}{38016} a^{14} - \frac{1}{144} a^{13} - \frac{1}{19008} a^{12} - \frac{5}{3456} a^{10} - \frac{1}{144} a^{9} + \frac{697}{19008} a^{8} + \frac{1}{36} a^{7} + \frac{163}{3456} a^{6} + \frac{5}{48} a^{5} - \frac{335}{4224} a^{4} + \frac{1}{9} a^{3} - \frac{461}{1056} a^{2} - \frac{5}{12} a - \frac{7}{32}$, $\frac{1}{114048} a^{17} - \frac{1}{38016} a^{15} + \frac{17}{19008} a^{13} + \frac{289}{114048} a^{11} + \frac{53}{6336} a^{9} + \frac{1811}{38016} a^{7} + \frac{27169}{114048} a^{5} - \frac{1}{2} a^{4} + \frac{3937}{9504} a^{3} + \frac{203}{1056} a$, $\frac{1}{425315947203658368} a^{18} - \frac{115951384127}{70885991200609728} a^{16} + \frac{9724514701643}{47257327467073152} a^{14} - \frac{1}{144} a^{13} + \frac{185803982867659}{32716611323358336} a^{12} - \frac{1289684349467383}{141771982401219456} a^{10} - \frac{1}{144} a^{9} + \frac{771577786913}{15752442489024384} a^{8} + \frac{1}{36} a^{7} + \frac{12423416614957}{308647276635456} a^{6} + \frac{5}{48} a^{5} + \frac{9015856031571695}{141771982401219456} a^{4} - \frac{7}{18} a^{3} + \frac{770017380950449}{2953582966692072} a^{2} + \frac{1}{12} a + \frac{195348758203}{3616263197664}$, $\frac{1}{425315947203658368} a^{19} - \frac{115951384127}{70885991200609728} a^{17} - \frac{220209091933}{47257327467073152} a^{15} - \frac{68933941998557}{32716611323358336} a^{13} - \frac{245488351141903}{141771982401219456} a^{11} + \frac{61030504667491}{5250814163008128} a^{9} - \frac{7386646384583}{308647276635456} a^{7} - \frac{48414923876329705}{141771982401219456} a^{5} - \frac{1109535416035415}{2953582966692072} a^{3} - \frac{1}{2} a^{2} - \frac{6013988837797}{13259631724768} a$

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

Class group and class number

$C_{11}\times C_{22}$, which has order $242$ (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:  \( \frac{2738059}{849098422656} a^{19} - \frac{340597}{283032807552} a^{17} - \frac{17847205}{23586067296} a^{15} - \frac{3064733465}{849098422656} a^{13} + \frac{426433309}{8844775236} a^{11} + \frac{5250371975}{10482696576} a^{9} + \frac{1462723422397}{849098422656} a^{7} + \frac{340060542553}{141516403776} a^{5} + \frac{23916665917}{23586067296} a^{3} + \frac{41190461}{39707184} a + \frac{1}{2} \) (order $6$)
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:  \( 231426336.202 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-489}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{163}) \), \(\Q(\sqrt{-3}, \sqrt{163})\), 5.1.3825936.1 x5, 10.0.28631509956043776.1, 10.0.43913358828288.1 x5, 10.2.9543836652014592.1 x5

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

Sibling fields

Degree 10 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$163$163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.1$x^{2} - 163$$2$$1$$1$$C_2$$[\ ]_{2}$