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Properties

Label 20.12.1069756574...3125.1
Degree $20$
Signature $[12, 4]$
Discriminant $5^{15}\cdot 19^{5}\cdot 1699^{5}$
Root discriminant $44.82$
Ramified primes $5, 19, 1699$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T771

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1759, 5371, 5579, -7970, 12895, -31838, -47697, 92368, 49188, -91579, -26077, 41938, 10148, -9566, -2937, 1114, 458, -53, -33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 33*x^18 - 53*x^17 + 458*x^16 + 1114*x^15 - 2937*x^14 - 9566*x^13 + 10148*x^12 + 41938*x^11 - 26077*x^10 - 91579*x^9 + 49188*x^8 + 92368*x^7 - 47697*x^6 - 31838*x^5 + 12895*x^4 - 7970*x^3 + 5579*x^2 + 5371*x - 1759)
 
gp: K = bnfinit(x^20 - 33*x^18 - 53*x^17 + 458*x^16 + 1114*x^15 - 2937*x^14 - 9566*x^13 + 10148*x^12 + 41938*x^11 - 26077*x^10 - 91579*x^9 + 49188*x^8 + 92368*x^7 - 47697*x^6 - 31838*x^5 + 12895*x^4 - 7970*x^3 + 5579*x^2 + 5371*x - 1759, 1)
 

Normalized defining polynomial

\( x^{20} - 33 x^{18} - 53 x^{17} + 458 x^{16} + 1114 x^{15} - 2937 x^{14} - 9566 x^{13} + 10148 x^{12} + 41938 x^{11} - 26077 x^{10} - 91579 x^{9} + 49188 x^{8} + 92368 x^{7} - 47697 x^{6} - 31838 x^{5} + 12895 x^{4} - 7970 x^{3} + 5579 x^{2} + 5371 x - 1759 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1069756574259831337445098876953125=5^{15}\cdot 19^{5}\cdot 1699^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.82$
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:  $5, 19, 1699$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} - \frac{7}{19} a^{17} - \frac{1}{19} a^{16} - \frac{3}{19} a^{15} + \frac{2}{19} a^{14} - \frac{8}{19} a^{13} - \frac{8}{19} a^{12} - \frac{7}{19} a^{11} - \frac{3}{19} a^{10} - \frac{7}{19} a^{9} - \frac{4}{19} a^{8} - \frac{4}{19} a^{7} - \frac{2}{19} a^{6} - \frac{4}{19} a^{5} - \frac{2}{19} a^{4} - \frac{7}{19} a^{3} + \frac{1}{19} a^{2} + \frac{8}{19} a - \frac{4}{19}$, $\frac{1}{765626256635696752800078211719305441884526981} a^{19} + \frac{19363728505624902650620284639647851735693551}{765626256635696752800078211719305441884526981} a^{18} + \frac{343729270725352556674930014548697956659560656}{765626256635696752800078211719305441884526981} a^{17} - \frac{220702880697908904244881544652656406415109553}{765626256635696752800078211719305441884526981} a^{16} + \frac{108133619564182304592584387198373916646433502}{765626256635696752800078211719305441884526981} a^{15} + \frac{185312759875958729586154971097101851636321266}{765626256635696752800078211719305441884526981} a^{14} - \frac{38117707810202357678937461457689411145307381}{765626256635696752800078211719305441884526981} a^{13} + \frac{35261729906037595080083583786633890279595453}{765626256635696752800078211719305441884526981} a^{12} - \frac{366646284011558052489535795720229618358807491}{765626256635696752800078211719305441884526981} a^{11} + \frac{324050786535949954608261062307175206203040733}{765626256635696752800078211719305441884526981} a^{10} - \frac{556521481023360294631047076857777958592852}{2370359927664695829102409324208375981066647} a^{9} - \frac{11400380453590628192944396349624914161678059}{45036838625629220752945777159959143640266293} a^{8} + \frac{297251129519535289134799209304004307933787945}{765626256635696752800078211719305441884526981} a^{7} + \frac{380887599896092203375750740570626594030994455}{765626256635696752800078211719305441884526981} a^{6} - \frac{318900593794797378634377551220664135896638058}{765626256635696752800078211719305441884526981} a^{5} + \frac{202085267776969125751343402969976348016825708}{765626256635696752800078211719305441884526981} a^{4} + \frac{215617708248907784882833293417467452747705547}{765626256635696752800078211719305441884526981} a^{3} + \frac{150736086520602434277950869149560466706493214}{765626256635696752800078211719305441884526981} a^{2} + \frac{136179914099391642494554917053178016212900191}{765626256635696752800078211719305441884526981} a + \frac{341250927539945704761357300946907946776535122}{765626256635696752800078211719305441884526981}$

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:  $15$
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:  \( 1856215114.59 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T771:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n771 are not computed
Character table for t20n771 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.3256446753125.1

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 $20$ $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ R $20$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
5Data not computed
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
1699Data not computed