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Properties

Label 20.4.21540389351...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{22}\cdot 13^{10}$
Root discriminant $36.87$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![922676, 316500, -1849060, -130120, 1560060, -32686, -779920, 26950, 279280, -6180, -78143, 930, 17285, -90, -2860, -4, 330, 0, -25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 25*x^18 + 330*x^16 - 4*x^15 - 2860*x^14 - 90*x^13 + 17285*x^12 + 930*x^11 - 78143*x^10 - 6180*x^9 + 279280*x^8 + 26950*x^7 - 779920*x^6 - 32686*x^5 + 1560060*x^4 - 130120*x^3 - 1849060*x^2 + 316500*x + 922676)
 
gp: K = bnfinit(x^20 - 25*x^18 + 330*x^16 - 4*x^15 - 2860*x^14 - 90*x^13 + 17285*x^12 + 930*x^11 - 78143*x^10 - 6180*x^9 + 279280*x^8 + 26950*x^7 - 779920*x^6 - 32686*x^5 + 1560060*x^4 - 130120*x^3 - 1849060*x^2 + 316500*x + 922676, 1)
 

Normalized defining polynomial

\( x^{20} - 25 x^{18} + 330 x^{16} - 4 x^{15} - 2860 x^{14} - 90 x^{13} + 17285 x^{12} + 930 x^{11} - 78143 x^{10} - 6180 x^{9} + 279280 x^{8} + 26950 x^{7} - 779920 x^{6} - 32686 x^{5} + 1560060 x^{4} - 130120 x^{3} - 1849060 x^{2} + 316500 x + 922676 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21540389351406250000000000000000=2^{16}\cdot 5^{22}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.87$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} - \frac{1}{14} a^{16} + \frac{3}{14} a^{13} + \frac{1}{14} a^{12} + \frac{1}{14} a^{11} - \frac{1}{7} a^{8} - \frac{1}{2} a^{7} - \frac{1}{14} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{326859901927957341835021602907723854782529234963398} a^{19} + \frac{678205950594972744945634573159013607992407949843}{23347135851996952988215828779123132484466373925957} a^{18} - \frac{34753227294550955460332453677206082118891196071759}{163429950963978670917510801453861927391264617481699} a^{17} + \frac{76324622395157839936514355921194680023775496380097}{326859901927957341835021602907723854782529234963398} a^{16} + \frac{3188186212516478466925318563750921567670252206703}{46694271703993905976431657558246264968932747851914} a^{15} + \frac{24264799528859102078628793862482943116237642673547}{163429950963978670917510801453861927391264617481699} a^{14} + \frac{410932884305610367208786713492905835292578660747}{326859901927957341835021602907723854782529234963398} a^{13} + \frac{633485903008152435172955790591796810567453325113}{163429950963978670917510801453861927391264617481699} a^{12} + \frac{17494094652447773368210979112885926679833198431669}{326859901927957341835021602907723854782529234963398} a^{11} - \frac{322242102870033729652276076787531050955053642341}{46694271703993905976431657558246264968932747851914} a^{10} + \frac{17323785782935637818883615843436553215430386006157}{326859901927957341835021602907723854782529234963398} a^{9} + \frac{5730331418079891694426290194788346918740226616681}{326859901927957341835021602907723854782529234963398} a^{8} + \frac{115958970702640224858708021339738742910493281452385}{326859901927957341835021602907723854782529234963398} a^{7} + \frac{1776263659577185544797768317926459636950203055413}{46694271703993905976431657558246264968932747851914} a^{6} - \frac{35064810322138475902473997614436481741947292434622}{163429950963978670917510801453861927391264617481699} a^{5} - \frac{6169756872628691277447828830727500440896693512579}{23347135851996952988215828779123132484466373925957} a^{4} - \frac{15334818690364463620399756918835924021087831074906}{163429950963978670917510801453861927391264617481699} a^{3} + \frac{11761333837277378285630884142814515030210693612104}{163429950963978670917510801453861927391264617481699} a^{2} - \frac{42027399520249032048183336533521534742520167602781}{163429950963978670917510801453861927391264617481699} a - \frac{37476689730954144027063108197301561581569454899}{651115342485970800468170523720565447773962619449}$

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:  $11$
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:  \( 89042310.19064918 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 5.1.50000.1, 10.2.12500000000.1, 10.2.928232500000000.3, 10.2.4641162500000000.2

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$