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Properties

Label 20.6.14672147998...9375.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,3^{10}\cdot 5^{10}\cdot 239^{9}$
Root discriminant $45.53$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![529, -6808, 35827, -109620, 218300, -293603, 258225, -119760, -24941, 92029, -80876, 40768, -8489, -4278, 5396, -3068, 1174, -336, 69, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 69*x^18 - 336*x^17 + 1174*x^16 - 3068*x^15 + 5396*x^14 - 4278*x^13 - 8489*x^12 + 40768*x^11 - 80876*x^10 + 92029*x^9 - 24941*x^8 - 119760*x^7 + 258225*x^6 - 293603*x^5 + 218300*x^4 - 109620*x^3 + 35827*x^2 - 6808*x + 529)
 
gp: K = bnfinit(x^20 - 10*x^19 + 69*x^18 - 336*x^17 + 1174*x^16 - 3068*x^15 + 5396*x^14 - 4278*x^13 - 8489*x^12 + 40768*x^11 - 80876*x^10 + 92029*x^9 - 24941*x^8 - 119760*x^7 + 258225*x^6 - 293603*x^5 + 218300*x^4 - 109620*x^3 + 35827*x^2 - 6808*x + 529, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 69 x^{18} - 336 x^{17} + 1174 x^{16} - 3068 x^{15} + 5396 x^{14} - 4278 x^{13} - 8489 x^{12} + 40768 x^{11} - 80876 x^{10} + 92029 x^{9} - 24941 x^{8} - 119760 x^{7} + 258225 x^{6} - 293603 x^{5} + 218300 x^{4} - 109620 x^{3} + 35827 x^{2} - 6808 x + 529 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1467214799841538367562738193359375=-\,3^{10}\cdot 5^{10}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.53$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{2}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{12} - \frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{4669028794904625} a^{18} - \frac{1}{518780977211625} a^{17} + \frac{3271436137391}{4669028794904625} a^{16} - \frac{1137890830388}{203001251952375} a^{15} - \frac{391348652915429}{4669028794904625} a^{14} + \frac{44002705855076}{518780977211625} a^{13} - \frac{4067430479882}{69686996938875} a^{12} - \frac{34983491225093}{4669028794904625} a^{11} - \frac{2039662422574646}{4669028794904625} a^{10} + \frac{509371130592514}{4669028794904625} a^{9} + \frac{2036787882926512}{4669028794904625} a^{8} - \frac{2392005207769}{37352230359237} a^{7} + \frac{583009502098231}{4669028794904625} a^{6} - \frac{616118235417193}{1556342931634875} a^{5} - \frac{2026714842167668}{4669028794904625} a^{4} - \frac{43868530555766}{103756195442325} a^{3} - \frac{548653936978676}{1556342931634875} a^{2} - \frac{146703530998556}{1556342931634875} a - \frac{57130075800958}{203001251952375}$, $\frac{1}{116725719872615625} a^{19} + \frac{1}{38908573290871875} a^{18} - \frac{5599563117748267}{116725719872615625} a^{17} + \frac{946891503530693}{116725719872615625} a^{16} + \frac{3029816513821183}{116725719872615625} a^{15} - \frac{499580735115563}{38908573290871875} a^{14} - \frac{10461117753498686}{116725719872615625} a^{13} + \frac{55524565218748054}{116725719872615625} a^{12} + \frac{32091348765018463}{116725719872615625} a^{11} + \frac{39532213670399662}{116725719872615625} a^{10} - \frac{5653836630043879}{23345143974523125} a^{9} + \frac{24142453944147019}{116725719872615625} a^{8} + \frac{56758570265223931}{116725719872615625} a^{7} - \frac{916074998831941}{4323174810096875} a^{6} + \frac{32755179980649809}{116725719872615625} a^{5} + \frac{3063318953417888}{38908573290871875} a^{4} + \frac{12410005846890769}{38908573290871875} a^{3} + \frac{113369861524634}{1691677099603125} a^{2} + \frac{72357318858331}{4669028794904625} a + \frac{123348533513618}{1691677099603125}$

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:  $12$
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:  \( 1148899592.28 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$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$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed