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Properties

Label 20.4.36006738474...5921.4
Degree $20$
Signature $[4, 8]$
Discriminant $17^{4}\cdot 401^{11}$
Root discriminant $47.62$
Ramified primes $17, 401$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-283113, 1427622, -3274691, 4679392, -4806625, 3871339, -2509850, 1280304, -555542, 236582, -82471, 5824, 16312, -12731, 5631, -1834, 516, -144, 40, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 40*x^18 - 144*x^17 + 516*x^16 - 1834*x^15 + 5631*x^14 - 12731*x^13 + 16312*x^12 + 5824*x^11 - 82471*x^10 + 236582*x^9 - 555542*x^8 + 1280304*x^7 - 2509850*x^6 + 3871339*x^5 - 4806625*x^4 + 4679392*x^3 - 3274691*x^2 + 1427622*x - 283113)
 
gp: K = bnfinit(x^20 - 9*x^19 + 40*x^18 - 144*x^17 + 516*x^16 - 1834*x^15 + 5631*x^14 - 12731*x^13 + 16312*x^12 + 5824*x^11 - 82471*x^10 + 236582*x^9 - 555542*x^8 + 1280304*x^7 - 2509850*x^6 + 3871339*x^5 - 4806625*x^4 + 4679392*x^3 - 3274691*x^2 + 1427622*x - 283113, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 40 x^{18} - 144 x^{17} + 516 x^{16} - 1834 x^{15} + 5631 x^{14} - 12731 x^{13} + 16312 x^{12} + 5824 x^{11} - 82471 x^{10} + 236582 x^{9} - 555542 x^{8} + 1280304 x^{7} - 2509850 x^{6} + 3871339 x^{5} - 4806625 x^{4} + 4679392 x^{3} - 3274691 x^{2} + 1427622 x - 283113 \)

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:  \(3600673847451055477688605112375921=17^{4}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.62$
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:  $17, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{2}{9} a^{11} - \frac{1}{9} a^{10} - \frac{4}{9} a^{8} + \frac{1}{3} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{4}{9} a^{11} - \frac{2}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{2}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{2}{9} a^{11} + \frac{1}{9} a^{10} - \frac{2}{9} a^{9} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{27} a^{15} + \frac{1}{27} a^{14} - \frac{1}{9} a^{13} + \frac{1}{27} a^{12} - \frac{2}{9} a^{11} - \frac{13}{27} a^{10} + \frac{2}{27} a^{9} - \frac{4}{27} a^{8} + \frac{8}{27} a^{7} + \frac{1}{27} a^{6} + \frac{1}{27} a^{5} - \frac{11}{27} a^{4} - \frac{11}{27} a^{3} - \frac{7}{27} a^{2} - \frac{1}{9} a$, $\frac{1}{143499186917287014830984391166099583916997400536609407} a^{19} + \frac{388806879925160920531618089798408177220703057501278}{143499186917287014830984391166099583916997400536609407} a^{18} + \frac{4375340680848080834275093111514406367431038938746712}{143499186917287014830984391166099583916997400536609407} a^{17} - \frac{7868013799702334529930835100657432117794269860270143}{143499186917287014830984391166099583916997400536609407} a^{16} - \frac{2785084051234328551753319330300010286695089598581008}{143499186917287014830984391166099583916997400536609407} a^{15} - \frac{13569691783164295070092452250478963273204810447652985}{143499186917287014830984391166099583916997400536609407} a^{14} + \frac{5478132460516063772866748874451006212134862253114075}{143499186917287014830984391166099583916997400536609407} a^{13} - \frac{2040891609843146723598288243538720854598862362286164}{143499186917287014830984391166099583916997400536609407} a^{12} + \frac{61053778191577911774194229716751318839572178871278}{143499186917287014830984391166099583916997400536609407} a^{11} + \frac{19170710974720261733914904473426132898122714244499677}{47833062305762338276994797055366527972332466845536469} a^{10} - \frac{295171811993047452816835921175084695977186983649299}{15944354101920779425664932351788842657444155615178823} a^{9} - \frac{19093965699481900698479126552511427513978982715860713}{47833062305762338276994797055366527972332466845536469} a^{8} - \frac{56340461518138035444199406953961679517542715903221775}{143499186917287014830984391166099583916997400536609407} a^{7} - \frac{5940213434742163220074097344884459419774892335663854}{15944354101920779425664932351788842657444155615178823} a^{6} + \frac{15090526718881648459169021608000085178072951675779366}{47833062305762338276994797055366527972332466845536469} a^{5} + \frac{6637646029485176042517300445182471867754103809491452}{47833062305762338276994797055366527972332466845536469} a^{4} + \frac{29852264877884771137222248042877145135314693851550462}{143499186917287014830984391166099583916997400536609407} a^{3} - \frac{17546503644017588995468069786768317867051278756542070}{143499186917287014830984391166099583916997400536609407} a^{2} + \frac{15329787385978008770983774149873223955693898999450656}{47833062305762338276994797055366527972332466845536469} a + \frac{5634849737511362992102084646768236548231441616902820}{15944354101920779425664932351788842657444155615178823}$

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:  \( 466770383.082 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n350 are not computed
Character table for t20n350 is not computed

Intermediate fields

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

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed