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Properties

Label 20.0.10326930237...0401.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 53^{10}$
Root discriminant $12.61$
Ramified primes $3, 53$
Class number $1$
Class group Trivial
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![1, -1, 24, 12, 57, 54, 52, -43, -32, -24, 23, 33, 19, 26, -29, -24, 6, 12, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 12*x^17 + 6*x^16 - 24*x^15 - 29*x^14 + 26*x^13 + 19*x^12 + 33*x^11 + 23*x^10 - 24*x^9 - 32*x^8 - 43*x^7 + 52*x^6 + 54*x^5 + 57*x^4 + 12*x^3 + 24*x^2 - x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 12*x^17 + 6*x^16 - 24*x^15 - 29*x^14 + 26*x^13 + 19*x^12 + 33*x^11 + 23*x^10 - 24*x^9 - 32*x^8 - 43*x^7 + 52*x^6 + 54*x^5 + 57*x^4 + 12*x^3 + 24*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 12 x^{17} + 6 x^{16} - 24 x^{15} - 29 x^{14} + 26 x^{13} + 19 x^{12} + 33 x^{11} + 23 x^{10} - 24 x^{9} - 32 x^{8} - 43 x^{7} + 52 x^{6} + 54 x^{5} + 57 x^{4} + 12 x^{3} + 24 x^{2} - x + 1 \)

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:  \(10326930237613180030401=3^{10}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.61$
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:  $3, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{13} - \frac{1}{3} a^{12} - \frac{1}{2} a^{11} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{18} a^{16} + \frac{1}{18} a^{15} - \frac{1}{18} a^{14} - \frac{1}{6} a^{13} + \frac{1}{18} a^{12} + \frac{1}{9} a^{11} - \frac{1}{6} a^{10} - \frac{1}{9} a^{9} - \frac{1}{2} a^{8} + \frac{1}{18} a^{7} + \frac{1}{6} a^{6} + \frac{5}{18} a^{5} + \frac{7}{18} a^{4} + \frac{1}{6} a^{3} - \frac{7}{18} a^{2} + \frac{2}{9} a + \frac{1}{18}$, $\frac{1}{18} a^{17} + \frac{1}{18} a^{15} - \frac{1}{9} a^{14} + \frac{1}{18} a^{13} - \frac{5}{18} a^{12} + \frac{2}{9} a^{11} - \frac{1}{9} a^{10} + \frac{5}{18} a^{9} - \frac{4}{9} a^{8} - \frac{7}{18} a^{7} - \frac{2}{9} a^{6} - \frac{1}{18} a^{5} - \frac{2}{9} a^{4} - \frac{7}{18} a^{3} - \frac{1}{18} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{54} a^{18} - \frac{1}{54} a^{16} - \frac{1}{54} a^{15} + \frac{1}{18} a^{14} - \frac{10}{27} a^{13} + \frac{7}{27} a^{12} - \frac{5}{18} a^{11} - \frac{5}{27} a^{10} + \frac{13}{27} a^{9} - \frac{7}{54} a^{8} - \frac{5}{18} a^{7} - \frac{13}{54} a^{6} - \frac{17}{54} a^{5} - \frac{7}{18} a^{4} - \frac{2}{27} a^{3} - \frac{5}{27} a^{2} - \frac{5}{18} a - \frac{17}{54}$, $\frac{1}{929837399216562} a^{19} - \frac{2423136176485}{464918699608281} a^{18} + \frac{9000832156121}{929837399216562} a^{17} - \frac{871487830111}{464918699608281} a^{16} - \frac{20313443307560}{464918699608281} a^{15} - \frac{56729948299625}{929837399216562} a^{14} + \frac{1485539480147}{103315266579618} a^{13} - \frac{21655996777316}{464918699608281} a^{12} - \frac{187049914568015}{464918699608281} a^{11} - \frac{319426505184359}{929837399216562} a^{10} + \frac{463242408992407}{929837399216562} a^{9} + \frac{117118392982747}{464918699608281} a^{8} + \frac{223089764428459}{464918699608281} a^{7} - \frac{940430106815}{17219211096603} a^{6} - \frac{142034519314561}{464918699608281} a^{5} - \frac{278880026568907}{929837399216562} a^{4} - \frac{395763205115135}{929837399216562} a^{3} - \frac{53329239500585}{464918699608281} a^{2} - \frac{357592090933115}{929837399216562} a - \frac{455944811078441}{929837399216562}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{263892301}{1021766778} a^{19} - \frac{512523166}{510883389} a^{18} - \frac{68770031}{510883389} a^{17} + \frac{3218511505}{1021766778} a^{16} + \frac{1976398211}{1021766778} a^{15} - \frac{3132890761}{510883389} a^{14} - \frac{1445557690}{170294463} a^{13} + \frac{3076463839}{510883389} a^{12} + \frac{6635284721}{1021766778} a^{11} + \frac{9516223105}{1021766778} a^{10} + \frac{3192926855}{510883389} a^{9} - \frac{6988428349}{1021766778} a^{8} - \frac{9422993593}{1021766778} a^{7} - \frac{4082735353}{340588926} a^{6} + \frac{13146663787}{1021766778} a^{5} + \frac{8621421553}{510883389} a^{4} + \frac{7880758943}{510883389} a^{3} + \frac{1696375360}{510883389} a^{2} + \frac{2357347160}{510883389} a + \frac{525802777}{1021766778} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1485.3179174 \)
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

\(\Q(\sqrt{-159}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{-3}, \sqrt{53})\), 5.1.25281.1 x5, 10.0.101621504799.2, 10.0.1917386883.1 x5, 10.2.33873834933.1 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$53$53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$