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Properties

Label 20.8.12569066605...8944.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{30}\cdot 11^{18}\cdot 1451^{2}$
Root discriminant $50.70$
Ramified primes $2, 11, 1451$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1376, -21248, 57888, 36768, -226768, 195552, 157856, -295200, 132672, 21824, -68936, 44176, -21108, 9552, -4416, 2008, -586, 104, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 + 104*x^17 - 586*x^16 + 2008*x^15 - 4416*x^14 + 9552*x^13 - 21108*x^12 + 44176*x^11 - 68936*x^10 + 21824*x^9 + 132672*x^8 - 295200*x^7 + 157856*x^6 + 195552*x^5 - 226768*x^4 + 36768*x^3 + 57888*x^2 - 21248*x + 1376)
 
gp: K = bnfinit(x^20 - 4*x^19 + 6*x^18 + 104*x^17 - 586*x^16 + 2008*x^15 - 4416*x^14 + 9552*x^13 - 21108*x^12 + 44176*x^11 - 68936*x^10 + 21824*x^9 + 132672*x^8 - 295200*x^7 + 157856*x^6 + 195552*x^5 - 226768*x^4 + 36768*x^3 + 57888*x^2 - 21248*x + 1376, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 6 x^{18} + 104 x^{17} - 586 x^{16} + 2008 x^{15} - 4416 x^{14} + 9552 x^{13} - 21108 x^{12} + 44176 x^{11} - 68936 x^{10} + 21824 x^{9} + 132672 x^{8} - 295200 x^{7} + 157856 x^{6} + 195552 x^{5} - 226768 x^{4} + 36768 x^{3} + 57888 x^{2} - 21248 x + 1376 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12569066605710630176407970532818944=2^{30}\cdot 11^{18}\cdot 1451^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.70$
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:  $2, 11, 1451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{16} a^{18}$, $\frac{1}{680414316199430070380639764280982005290451951728} a^{19} + \frac{6530610969633766766933081556601592717841846193}{340207158099715035190319882140491002645225975864} a^{18} + \frac{9179003389788542310328347590391833960696440677}{680414316199430070380639764280982005290451951728} a^{17} - \frac{517029741316076629668697957752642510082552155}{85051789524928758797579970535122750661306493966} a^{16} + \frac{2152656570440039013253309117865658102656777587}{170103579049857517595159941070245501322612987932} a^{15} - \frac{7769462350250587941409687537583566921588977159}{340207158099715035190319882140491002645225975864} a^{14} + \frac{3151171229063548383362011044492315914088851461}{85051789524928758797579970535122750661306493966} a^{13} + \frac{1154945217091201613061750211707885431491718237}{42525894762464379398789985267561375330653246983} a^{12} - \frac{7257954785743679295634127582905353899758155569}{170103579049857517595159941070245501322612987932} a^{11} - \frac{9108695485663115255454658815104932759657099527}{85051789524928758797579970535122750661306493966} a^{10} - \frac{259944347778218965867755767288458595018118383}{3955897187205988781282789327215011658665418324} a^{9} + \frac{3714940393184264293276573637924034766798974642}{42525894762464379398789985267561375330653246983} a^{8} - \frac{16900794530756693780987092875144617145053212019}{85051789524928758797579970535122750661306493966} a^{7} - \frac{6657767828447853201743162622963342122037513801}{85051789524928758797579970535122750661306493966} a^{6} - \frac{17934197476005188970258044992447031658567241193}{85051789524928758797579970535122750661306493966} a^{5} - \frac{8066699803770983766029148908590127410739319019}{42525894762464379398789985267561375330653246983} a^{4} - \frac{15869059766763266025065784479840790955932815355}{42525894762464379398789985267561375330653246983} a^{3} + \frac{3956413599568451421298077914746937225901266134}{42525894762464379398789985267561375330653246983} a^{2} - \frac{19130523305595456124206507505101517152430142869}{42525894762464379398789985267561375330653246983} a + \frac{338871141517276103358266556643457506045337768}{988974296801497195320697331803752914666354581}$

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:  $13$
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:  \( 1384651748.02 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1451Data not computed