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Properties

Label 20.6.14113234290...1603.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,13^{12}\cdot 347^{7}$
Root discriminant $36.10$
Ramified primes $13, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59959, 7671, -102634, 50946, -28829, -44088, 72856, -59822, 31475, -5961, -6505, 7740, -2979, -415, 862, -241, -79, 63, -10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 10*x^18 + 63*x^17 - 79*x^16 - 241*x^15 + 862*x^14 - 415*x^13 - 2979*x^12 + 7740*x^11 - 6505*x^10 - 5961*x^9 + 31475*x^8 - 59822*x^7 + 72856*x^6 - 44088*x^5 - 28829*x^4 + 50946*x^3 - 102634*x^2 + 7671*x + 59959)
 
gp: K = bnfinit(x^20 - 3*x^19 - 10*x^18 + 63*x^17 - 79*x^16 - 241*x^15 + 862*x^14 - 415*x^13 - 2979*x^12 + 7740*x^11 - 6505*x^10 - 5961*x^9 + 31475*x^8 - 59822*x^7 + 72856*x^6 - 44088*x^5 - 28829*x^4 + 50946*x^3 - 102634*x^2 + 7671*x + 59959, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 10 x^{18} + 63 x^{17} - 79 x^{16} - 241 x^{15} + 862 x^{14} - 415 x^{13} - 2979 x^{12} + 7740 x^{11} - 6505 x^{10} - 5961 x^{9} + 31475 x^{8} - 59822 x^{7} + 72856 x^{6} - 44088 x^{5} - 28829 x^{4} + 50946 x^{3} - 102634 x^{2} + 7671 x + 59959 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-14113234290632913459536084681603=-\,13^{12}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.10$
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:  $13, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{59} a^{18} + \frac{8}{59} a^{17} + \frac{2}{59} a^{16} + \frac{8}{59} a^{15} - \frac{25}{59} a^{14} - \frac{3}{59} a^{13} + \frac{15}{59} a^{12} - \frac{22}{59} a^{11} + \frac{5}{59} a^{10} + \frac{27}{59} a^{9} + \frac{20}{59} a^{8} - \frac{5}{59} a^{7} - \frac{13}{59} a^{6} + \frac{5}{59} a^{5} - \frac{28}{59} a^{4} + \frac{5}{59} a^{3} + \frac{22}{59} a^{2} + \frac{9}{59} a - \frac{13}{59}$, $\frac{1}{1194025598452395625382737868861108040033345789569} a^{19} - \frac{9633027278887532668715161650135038033627136389}{1194025598452395625382737868861108040033345789569} a^{18} - \frac{344250723862711208389953959784770602470865426228}{1194025598452395625382737868861108040033345789569} a^{17} - \frac{273558100834532567369409567993944114478046361724}{1194025598452395625382737868861108040033345789569} a^{16} + \frac{511476410423596848053060814314044902824617692461}{1194025598452395625382737868861108040033345789569} a^{15} - \frac{568454262397855197168239499509012452584944010888}{1194025598452395625382737868861108040033345789569} a^{14} + \frac{71055410960384817452471593202473293759350962995}{1194025598452395625382737868861108040033345789569} a^{13} - \frac{435699313618048189152027012561359002872517676964}{1194025598452395625382737868861108040033345789569} a^{12} - \frac{575054519323036594951891721232896467865751388917}{1194025598452395625382737868861108040033345789569} a^{11} + \frac{172456923988632211000535066858138682122323602978}{1194025598452395625382737868861108040033345789569} a^{10} - \frac{289765230443832476829696977995671755191931249119}{1194025598452395625382737868861108040033345789569} a^{9} + \frac{170645732828137254469219197570437966859328615050}{1194025598452395625382737868861108040033345789569} a^{8} - \frac{119693066189414816853036101336037223498976050890}{1194025598452395625382737868861108040033345789569} a^{7} + \frac{470732970684477079683769166165035158157671219111}{1194025598452395625382737868861108040033345789569} a^{6} + \frac{63428652562610705605961936380711143642530736095}{1194025598452395625382737868861108040033345789569} a^{5} + \frac{500743623626334110248974126567325627369052124404}{1194025598452395625382737868861108040033345789569} a^{4} - \frac{390990018766643020285342537935222309457642657056}{1194025598452395625382737868861108040033345789569} a^{3} - \frac{93621443225770447114664508784984641656839516792}{1194025598452395625382737868861108040033345789569} a^{2} - \frac{463937360792141747363394912881232761818415716609}{1194025598452395625382737868861108040033345789569} a - \frac{398346568875814229583426773693924292880625902659}{1194025598452395625382737868861108040033345789569}$

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:  $12$
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:  \( 50653808.4647 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
347Data not computed