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Properties

Label 20.10.4074580816...6379.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,379\cdot 401^{10}$
Root discriminant $26.95$
Ramified primes $379, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 10, 28, -11, -41, -193, 80, 273, 123, -29, -280, 232, 39, -198, 68, -9, -5, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 6*x^18 - 5*x^17 - 9*x^16 + 68*x^15 - 198*x^14 + 39*x^13 + 232*x^12 - 280*x^11 - 29*x^10 + 123*x^9 + 273*x^8 + 80*x^7 - 193*x^6 - 41*x^5 - 11*x^4 + 28*x^3 + 10*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 6*x^18 - 5*x^17 - 9*x^16 + 68*x^15 - 198*x^14 + 39*x^13 + 232*x^12 - 280*x^11 - 29*x^10 + 123*x^9 + 273*x^8 + 80*x^7 - 193*x^6 - 41*x^5 - 11*x^4 + 28*x^3 + 10*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 6 x^{18} - 5 x^{17} - 9 x^{16} + 68 x^{15} - 198 x^{14} + 39 x^{13} + 232 x^{12} - 280 x^{11} - 29 x^{10} + 123 x^{9} + 273 x^{8} + 80 x^{7} - 193 x^{6} - 41 x^{5} - 11 x^{4} + 28 x^{3} + 10 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-40745808166212682337450316379=-\,379\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.95$
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:  $379, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{27} a^{18} - \frac{1}{27} a^{17} + \frac{2}{27} a^{16} + \frac{2}{27} a^{15} - \frac{5}{27} a^{14} - \frac{1}{9} a^{13} - \frac{13}{27} a^{12} + \frac{1}{9} a^{11} + \frac{11}{27} a^{10} - \frac{7}{27} a^{9} - \frac{7}{27} a^{8} + \frac{1}{27} a^{7} + \frac{13}{27} a^{6} + \frac{10}{27} a^{5} + \frac{13}{27} a^{4} - \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{5}{27} a + \frac{1}{27}$, $\frac{1}{4058299027795630677831} a^{19} - \frac{70067373266252077345}{4058299027795630677831} a^{18} - \frac{625295843711844768511}{4058299027795630677831} a^{17} - \frac{491343993236715603523}{4058299027795630677831} a^{16} + \frac{1757998298817671285407}{4058299027795630677831} a^{15} + \frac{471327305414765800562}{1352766342598543559277} a^{14} - \frac{233901996673325554273}{4058299027795630677831} a^{13} - \frac{465667999696597662605}{1352766342598543559277} a^{12} + \frac{1909947745175073346001}{4058299027795630677831} a^{11} - \frac{190922006386263258763}{4058299027795630677831} a^{10} - \frac{977353537325656319494}{4058299027795630677831} a^{9} + \frac{828890940918962531026}{4058299027795630677831} a^{8} - \frac{1594249749701796276626}{4058299027795630677831} a^{7} - \frac{968057858452512103721}{4058299027795630677831} a^{6} + \frac{1495346989471828032352}{4058299027795630677831} a^{5} - \frac{142317789058880653561}{1352766342598543559277} a^{4} - \frac{314763239527903176407}{1352766342598543559277} a^{3} - \frac{11967845117452350629}{4058299027795630677831} a^{2} - \frac{1190061558211604382206}{4058299027795630677831} a + \frac{5266428123628228668}{150307371399838173253}$

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:  $14$
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:  \( 9211561.23937 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
379Data not computed
401Data not computed