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Properties

Label 20.0.16515022032...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{27}\cdot 5^{14}\cdot 17^{10}$
Root discriminant $32.43$
Ramified primes $2, 5, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1226, -274, 3946, -560, -1860, 5360, 1808, -6760, 7072, -1404, -2237, 2129, -203, -646, 401, -25, -49, 10, 9, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 9*x^18 + 10*x^17 - 49*x^16 - 25*x^15 + 401*x^14 - 646*x^13 - 203*x^12 + 2129*x^11 - 2237*x^10 - 1404*x^9 + 7072*x^8 - 6760*x^7 + 1808*x^6 + 5360*x^5 - 1860*x^4 - 560*x^3 + 3946*x^2 - 274*x + 1226)
 
gp: K = bnfinit(x^20 - 5*x^19 + 9*x^18 + 10*x^17 - 49*x^16 - 25*x^15 + 401*x^14 - 646*x^13 - 203*x^12 + 2129*x^11 - 2237*x^10 - 1404*x^9 + 7072*x^8 - 6760*x^7 + 1808*x^6 + 5360*x^5 - 1860*x^4 - 560*x^3 + 3946*x^2 - 274*x + 1226, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 9 x^{18} + 10 x^{17} - 49 x^{16} - 25 x^{15} + 401 x^{14} - 646 x^{13} - 203 x^{12} + 2129 x^{11} - 2237 x^{10} - 1404 x^{9} + 7072 x^{8} - 6760 x^{7} + 1808 x^{6} + 5360 x^{5} - 1860 x^{4} - 560 x^{3} + 3946 x^{2} - 274 x + 1226 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1651502203247820800000000000000=2^{27}\cdot 5^{14}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.43$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3209} a^{18} + \frac{404}{3209} a^{17} - \frac{1162}{3209} a^{16} - \frac{194}{3209} a^{15} + \frac{1051}{3209} a^{14} + \frac{246}{3209} a^{13} + \frac{1488}{3209} a^{12} - \frac{673}{3209} a^{11} - \frac{244}{3209} a^{10} - \frac{377}{3209} a^{9} + \frac{642}{3209} a^{8} + \frac{734}{3209} a^{7} - \frac{52}{3209} a^{6} - \frac{927}{3209} a^{5} - \frac{182}{3209} a^{4} + \frac{970}{3209} a^{3} - \frac{305}{3209} a^{2} + \frac{965}{3209} a + \frac{1306}{3209}$, $\frac{1}{1066397840372815142201506398908123819} a^{19} - \frac{54115051266941030428307258192077}{1066397840372815142201506398908123819} a^{18} - \frac{115903480530100543432153193624269991}{1066397840372815142201506398908123819} a^{17} + \frac{423224542213297672716071647069972066}{1066397840372815142201506398908123819} a^{16} + \frac{324834692104213814127398060689265332}{1066397840372815142201506398908123819} a^{15} - \frac{158478726059424150530255444565665505}{1066397840372815142201506398908123819} a^{14} + \frac{262238678761492461758138811875845211}{1066397840372815142201506398908123819} a^{13} + \frac{164366746246189575107117665218131222}{1066397840372815142201506398908123819} a^{12} + \frac{163383656164087077704567299581984248}{1066397840372815142201506398908123819} a^{11} - \frac{54422682281689094926659573795296965}{1066397840372815142201506398908123819} a^{10} + \frac{326471585685815387781618088849202838}{1066397840372815142201506398908123819} a^{9} + \frac{228707851646973256628877982493350663}{1066397840372815142201506398908123819} a^{8} - \frac{328914033275126010872282383285905093}{1066397840372815142201506398908123819} a^{7} - \frac{225820036046642777524754223152863616}{1066397840372815142201506398908123819} a^{6} - \frac{500958800551443592953270088632443372}{1066397840372815142201506398908123819} a^{5} + \frac{77946176665516277977618504316951004}{1066397840372815142201506398908123819} a^{4} + \frac{185610039844142450725612356409927624}{1066397840372815142201506398908123819} a^{3} - \frac{398072425502826969658012773617422237}{1066397840372815142201506398908123819} a^{2} - \frac{38186712938375246795781338876798903}{1066397840372815142201506398908123819} a + \frac{348865639186283469962665687582461941}{1066397840372815142201506398908123819}$

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:  $9$
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:  \( 9169008.40398 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.57800.1, 5.1.578000.2, 10.2.5679428000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.10.19.17$x^{10} - 2 x^{4} + 4 x^{2} - 10$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$