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Properties

Label 20.20.1853091914...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{14}\cdot 6029^{7}$
Root discriminant $129.84$
Ramified primes $2, 5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:S_5$ (as 20T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![219147162389, 0, -485802259965, 0, 291125849965, 0, -79010543635, 0, 11493603485, 0, -970837684, 0, 49256545, 0, -1503545, 0, 26780, 0, -255, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 255*x^18 + 26780*x^16 - 1503545*x^14 + 49256545*x^12 - 970837684*x^10 + 11493603485*x^8 - 79010543635*x^6 + 291125849965*x^4 - 485802259965*x^2 + 219147162389)
 
gp: K = bnfinit(x^20 - 255*x^18 + 26780*x^16 - 1503545*x^14 + 49256545*x^12 - 970837684*x^10 + 11493603485*x^8 - 79010543635*x^6 + 291125849965*x^4 - 485802259965*x^2 + 219147162389, 1)
 

Normalized defining polynomial

\( x^{20} - 255 x^{18} + 26780 x^{16} - 1503545 x^{14} + 49256545 x^{12} - 970837684 x^{10} + 11493603485 x^{8} - 79010543635 x^{6} + 291125849965 x^{4} - 485802259965 x^{2} + 219147162389 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1853091914135141046229493177600000000000000=2^{20}\cdot 5^{14}\cdot 6029^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.84$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{25} a^{8} + \frac{2}{25} a^{6} - \frac{1}{25} a^{4} - \frac{2}{25} a^{2} + \frac{1}{25}$, $\frac{1}{25} a^{9} + \frac{2}{25} a^{7} - \frac{1}{25} a^{5} - \frac{2}{25} a^{3} + \frac{1}{25} a$, $\frac{1}{25} a^{10} - \frac{1}{5} a^{2} + \frac{3}{25}$, $\frac{1}{25} a^{11} - \frac{1}{5} a^{3} + \frac{3}{25} a$, $\frac{1}{125} a^{12} - \frac{2}{125} a^{10} - \frac{1}{25} a^{6} + \frac{28}{125} a^{2} - \frac{16}{125}$, $\frac{1}{125} a^{13} - \frac{2}{125} a^{11} - \frac{1}{25} a^{7} + \frac{28}{125} a^{3} - \frac{16}{125} a$, $\frac{1}{625} a^{14} + \frac{1}{625} a^{12} - \frac{1}{625} a^{10} - \frac{1}{125} a^{8} + \frac{2}{125} a^{6} - \frac{22}{625} a^{4} + \frac{193}{625} a^{2} + \frac{167}{625}$, $\frac{1}{625} a^{15} + \frac{1}{625} a^{13} - \frac{1}{625} a^{11} - \frac{1}{125} a^{9} + \frac{2}{125} a^{7} - \frac{22}{625} a^{5} + \frac{193}{625} a^{3} + \frac{167}{625} a$, $\frac{1}{48985625} a^{16} + \frac{5978}{9797125} a^{14} + \frac{68983}{48985625} a^{12} - \frac{243484}{48985625} a^{10} + \frac{120503}{9797125} a^{8} - \frac{4461332}{48985625} a^{6} + \frac{74012}{9797125} a^{4} - \frac{3718121}{48985625} a^{2} - \frac{4058}{8125}$, $\frac{1}{48985625} a^{17} + \frac{5978}{9797125} a^{15} + \frac{68983}{48985625} a^{13} - \frac{243484}{48985625} a^{11} + \frac{120503}{9797125} a^{9} - \frac{4461332}{48985625} a^{7} + \frac{74012}{9797125} a^{5} - \frac{3718121}{48985625} a^{3} - \frac{4058}{8125} a$, $\frac{1}{955239799887892453713685894365625} a^{18} - \frac{4277452598392926254812378}{955239799887892453713685894365625} a^{16} + \frac{364021332761465472652993809039}{955239799887892453713685894365625} a^{14} - \frac{3183518301596805736029661557582}{955239799887892453713685894365625} a^{12} - \frac{1765478382566083044804592089639}{955239799887892453713685894365625} a^{10} - \frac{2382995963067169810190395999157}{955239799887892453713685894365625} a^{8} - \frac{84829657499240209379507650926654}{955239799887892453713685894365625} a^{6} - \frac{50725187682611781580385218951948}{955239799887892453713685894365625} a^{4} + \frac{55793028945353250735986720231}{158440835940934226855811228125} a^{2} - \frac{7437901897635173258891797}{26279787019561158874740625}$, $\frac{1}{955239799887892453713685894365625} a^{19} - \frac{4277452598392926254812378}{955239799887892453713685894365625} a^{17} + \frac{364021332761465472652993809039}{955239799887892453713685894365625} a^{15} - \frac{3183518301596805736029661557582}{955239799887892453713685894365625} a^{13} - \frac{1765478382566083044804592089639}{955239799887892453713685894365625} a^{11} - \frac{2382995963067169810190395999157}{955239799887892453713685894365625} a^{9} - \frac{84829657499240209379507650926654}{955239799887892453713685894365625} a^{7} - \frac{50725187682611781580385218951948}{955239799887892453713685894365625} a^{5} + \frac{55793028945353250735986720231}{158440835940934226855811228125} a^{3} - \frac{7437901897635173258891797}{26279787019561158874740625} a$

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:  $19$
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:  \( 481489568427000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:S_5$ (as 20T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 19 conjugacy class representatives for $C_4:S_5$
Character table for $C_4:S_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2411600.1, 5.5.753625.1, 10.10.2839753203125.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed