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Properties

Label 20.4.30705567995...2561.1
Degree $20$
Signature $[4, 8]$
Discriminant $13^{4}\cdot 401^{10}$
Root discriminant $33.45$
Ramified primes $13, 401$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4:D_5$ (as 20T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2313, 2208, -6098, 17249, -16095, 7654, 9051, -22354, 25785, -22104, 14732, -8556, 4356, -1955, 661, -108, -50, 35, 4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 4*x^18 + 35*x^17 - 50*x^16 - 108*x^15 + 661*x^14 - 1955*x^13 + 4356*x^12 - 8556*x^11 + 14732*x^10 - 22104*x^9 + 25785*x^8 - 22354*x^7 + 9051*x^6 + 7654*x^5 - 16095*x^4 + 17249*x^3 - 6098*x^2 + 2208*x + 2313)
 
gp: K = bnfinit(x^20 - 6*x^19 + 4*x^18 + 35*x^17 - 50*x^16 - 108*x^15 + 661*x^14 - 1955*x^13 + 4356*x^12 - 8556*x^11 + 14732*x^10 - 22104*x^9 + 25785*x^8 - 22354*x^7 + 9051*x^6 + 7654*x^5 - 16095*x^4 + 17249*x^3 - 6098*x^2 + 2208*x + 2313, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 4 x^{18} + 35 x^{17} - 50 x^{16} - 108 x^{15} + 661 x^{14} - 1955 x^{13} + 4356 x^{12} - 8556 x^{11} + 14732 x^{10} - 22104 x^{9} + 25785 x^{8} - 22354 x^{7} + 9051 x^{6} + 7654 x^{5} - 16095 x^{4} + 17249 x^{3} - 6098 x^{2} + 2208 x + 2313 \)

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:  \(3070556799565172612770233472561=13^{4}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.45$
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:  $13, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{14} + \frac{1}{6} a^{12} - \frac{1}{3} a^{11} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{114} a^{17} + \frac{1}{38} a^{16} + \frac{25}{114} a^{15} + \frac{9}{38} a^{14} + \frac{25}{114} a^{13} - \frac{5}{114} a^{12} + \frac{1}{6} a^{11} + \frac{53}{114} a^{10} + \frac{37}{114} a^{9} - \frac{29}{114} a^{8} + \frac{2}{57} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{5}{38} a^{4} - \frac{7}{57} a^{3} + \frac{28}{57} a^{2} - \frac{9}{38} a + \frac{5}{38}$, $\frac{1}{4446} a^{18} + \frac{4}{2223} a^{17} - \frac{94}{2223} a^{16} - \frac{2}{117} a^{15} - \frac{490}{2223} a^{14} + \frac{108}{247} a^{13} - \frac{229}{741} a^{12} + \frac{359}{2223} a^{11} + \frac{1006}{2223} a^{10} + \frac{235}{741} a^{9} - \frac{655}{1482} a^{8} + \frac{469}{1482} a^{7} + \frac{3}{26} a^{6} - \frac{1459}{4446} a^{5} + \frac{1621}{4446} a^{4} - \frac{919}{2223} a^{3} - \frac{545}{4446} a^{2} - \frac{191}{741} a - \frac{245}{494}$, $\frac{1}{391200062925414453934961011744100826} a^{19} - \frac{19579424174742070090527165731656}{195600031462707226967480505872050413} a^{18} - \frac{673690506383855865358807233511837}{391200062925414453934961011744100826} a^{17} - \frac{21878347837110423310347944619547115}{391200062925414453934961011744100826} a^{16} - \frac{30245533192523773641363886535022626}{195600031462707226967480505872050413} a^{15} - \frac{25588423152165904097137872279976991}{195600031462707226967480505872050413} a^{14} + \frac{29622729366725086243722813424490141}{130400020975138151311653670581366942} a^{13} + \frac{124664194400151998464901504266704115}{391200062925414453934961011744100826} a^{12} - \frac{644348989075578095709901542762124}{15046156266362094382113885067080801} a^{11} - \frac{60023371277674921600785945335781679}{195600031462707226967480505872050413} a^{10} + \frac{3780496003378180780341587601440797}{65200010487569075655826835290683471} a^{9} + \frac{14220015319168041053277457140527014}{65200010487569075655826835290683471} a^{8} - \frac{26301818767831529958028174053428210}{65200010487569075655826835290683471} a^{7} - \frac{116874698641996694354493805657547947}{391200062925414453934961011744100826} a^{6} + \frac{4891264397667926985204673312519792}{10294738498037222471972658203792127} a^{5} + \frac{14472903155831901246186263659862743}{130400020975138151311653670581366942} a^{4} - \frac{1220280261554336250254297476358780}{7244445609729897295091870587853719} a^{3} - \frac{44869650279108731898879880913792521}{391200062925414453934961011744100826} a^{2} - \frac{26536786756020855799314708703318363}{130400020975138151311653670581366942} a + \frac{5765472563440441992648179074031357}{21733336829189691885275611763561157}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 14458688.8061 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.2.4369826510569.1, 10.10.10368641602001.1, 10.2.1752300430738169.1

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
Degree 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed