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Properties

Label 20.8.18358429263...0000.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{6}$
Root discriminant $41.04$
Ramified primes $2, 5, 13, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -27, 279, 35, -5419, 17486, -17818, -15670, 33795, -16779, 1351, 5171, -4019, 2140, -962, 226, 9, 11, -7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 7*x^18 + 11*x^17 + 9*x^16 + 226*x^15 - 962*x^14 + 2140*x^13 - 4019*x^12 + 5171*x^11 + 1351*x^10 - 16779*x^9 + 33795*x^8 - 15670*x^7 - 17818*x^6 + 17486*x^5 - 5419*x^4 + 35*x^3 + 279*x^2 - 27*x + 1)
 
gp: K = bnfinit(x^20 - 3*x^19 - 7*x^18 + 11*x^17 + 9*x^16 + 226*x^15 - 962*x^14 + 2140*x^13 - 4019*x^12 + 5171*x^11 + 1351*x^10 - 16779*x^9 + 33795*x^8 - 15670*x^7 - 17818*x^6 + 17486*x^5 - 5419*x^4 + 35*x^3 + 279*x^2 - 27*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 7 x^{18} + 11 x^{17} + 9 x^{16} + 226 x^{15} - 962 x^{14} + 2140 x^{13} - 4019 x^{12} + 5171 x^{11} + 1351 x^{10} - 16779 x^{9} + 33795 x^{8} - 15670 x^{7} - 17818 x^{6} + 17486 x^{5} - 5419 x^{4} + 35 x^{3} + 279 x^{2} - 27 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(183584292633496284375040000000000=2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.04$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43} a^{18} + \frac{8}{43} a^{17} - \frac{2}{43} a^{16} + \frac{13}{43} a^{15} + \frac{17}{43} a^{14} - \frac{21}{43} a^{13} + \frac{19}{43} a^{12} + \frac{7}{43} a^{11} - \frac{15}{43} a^{10} - \frac{4}{43} a^{9} + \frac{15}{43} a^{8} + \frac{15}{43} a^{7} - \frac{8}{43} a^{6} - \frac{18}{43} a^{5} + \frac{20}{43} a^{4} - \frac{21}{43} a^{3} + \frac{15}{43} a + \frac{14}{43}$, $\frac{1}{456268889392888183983282396622004337684001} a^{19} - \frac{2640651653661097228402235056938934976283}{456268889392888183983282396622004337684001} a^{18} + \frac{71903945899149461037402979903321782252551}{456268889392888183983282396622004337684001} a^{17} - \frac{210358578555429285511187293018035224823196}{456268889392888183983282396622004337684001} a^{16} + \frac{187579852506851061808948430957107962579898}{456268889392888183983282396622004337684001} a^{15} + \frac{210813853517515270645918905197591360901342}{456268889392888183983282396622004337684001} a^{14} - \frac{38759879113381442978827507184105392265338}{456268889392888183983282396622004337684001} a^{13} + \frac{132817828857031099250419533547034511439088}{456268889392888183983282396622004337684001} a^{12} - \frac{209345147795622932866490641732036696989183}{456268889392888183983282396622004337684001} a^{11} - \frac{185766014021688604083604840987046772902386}{456268889392888183983282396622004337684001} a^{10} + \frac{219912733362553517835428751930464274123704}{456268889392888183983282396622004337684001} a^{9} + \frac{167150810143001343277739679068951985092369}{456268889392888183983282396622004337684001} a^{8} + \frac{100761237963063356862771706027247107126590}{456268889392888183983282396622004337684001} a^{7} + \frac{194044082616262235576994306804408569569015}{456268889392888183983282396622004337684001} a^{6} + \frac{190986569097548690184504563107723085094605}{456268889392888183983282396622004337684001} a^{5} + \frac{115637515166480625515960636477397052394721}{456268889392888183983282396622004337684001} a^{4} - \frac{129314279078234611256904888007635947751607}{456268889392888183983282396622004337684001} a^{3} + \frac{30759483464346949419721905412563952960447}{456268889392888183983282396622004337684001} a^{2} - \frac{197597151249604097915553600190935368951574}{456268889392888183983282396622004337684001} a + \frac{22584262602896799685904794502874115323740}{456268889392888183983282396622004337684001}$

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:  $13$
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:  \( 411620294.282 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.10.0.1$x^{10} + x^{2} - 2 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.6.5.4$x^{6} + 217$$6$$1$$5$$C_6$$[\ ]_{6}$