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Properties

Label 20.8.19481352064...8125.2
Degree $20$
Signature $[8, 6]$
Discriminant $3^{8}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $58.14$
Ramified primes $3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-711, -12933, -6090, 2772, -39049, -16878, 84333, 72007, -55089, -92095, -924, 49505, 12282, -11371, -2747, 1115, -107, -34, 39, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 39*x^18 - 34*x^17 - 107*x^16 + 1115*x^15 - 2747*x^14 - 11371*x^13 + 12282*x^12 + 49505*x^11 - 924*x^10 - 92095*x^9 - 55089*x^8 + 72007*x^7 + 84333*x^6 - 16878*x^5 - 39049*x^4 + 2772*x^3 - 6090*x^2 - 12933*x - 711)
 
gp: K = bnfinit(x^20 - 5*x^19 + 39*x^18 - 34*x^17 - 107*x^16 + 1115*x^15 - 2747*x^14 - 11371*x^13 + 12282*x^12 + 49505*x^11 - 924*x^10 - 92095*x^9 - 55089*x^8 + 72007*x^7 + 84333*x^6 - 16878*x^5 - 39049*x^4 + 2772*x^3 - 6090*x^2 - 12933*x - 711, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 39 x^{18} - 34 x^{17} - 107 x^{16} + 1115 x^{15} - 2747 x^{14} - 11371 x^{13} + 12282 x^{12} + 49505 x^{11} - 924 x^{10} - 92095 x^{9} - 55089 x^{8} + 72007 x^{7} + 84333 x^{6} - 16878 x^{5} - 39049 x^{4} + 2772 x^{3} - 6090 x^{2} - 12933 x - 711 \)

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:  \(194813520645626483248608015673828125=3^{8}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.14$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{1611} a^{18} + \frac{214}{1611} a^{17} - \frac{43}{537} a^{16} - \frac{109}{1611} a^{15} + \frac{10}{1611} a^{14} + \frac{587}{1611} a^{13} + \frac{220}{1611} a^{12} + \frac{116}{1611} a^{11} + \frac{202}{537} a^{10} + \frac{689}{1611} a^{9} - \frac{72}{179} a^{8} - \frac{499}{1611} a^{7} + \frac{61}{179} a^{6} - \frac{164}{1611} a^{5} - \frac{157}{537} a^{4} + \frac{41}{179} a^{3} + \frac{566}{1611} a^{2} + \frac{32}{179} a + \frac{197}{537}$, $\frac{1}{2070113138601208678295936907053690503945744317} a^{19} - \frac{262782504019608242738960565504063287098834}{2070113138601208678295936907053690503945744317} a^{18} + \frac{41924994649020378748203049984434943173914389}{2070113138601208678295936907053690503945744317} a^{17} + \frac{153947786623609614214923167928223694132530088}{2070113138601208678295936907053690503945744317} a^{16} - \frac{70879614444779427241655867599843907080877686}{690037712867069559431978969017896834648581439} a^{15} - \frac{98876355880511203823269669553154336045527788}{690037712867069559431978969017896834648581439} a^{14} - \frac{79157285780067713350593708188754733967856764}{230012570955689853143992989672632278216193813} a^{13} - \frac{58502416164858326904534492788215741990281049}{230012570955689853143992989672632278216193813} a^{12} + \frac{850542891571994010274786988159791364177240624}{2070113138601208678295936907053690503945744317} a^{11} - \frac{740941675482080082543158800797193822723633300}{2070113138601208678295936907053690503945744317} a^{10} + \frac{221520296781782946561578780828725287209691872}{2070113138601208678295936907053690503945744317} a^{9} + \frac{21602698269069043293402593679205505651781820}{71383211675903747527446100243230707032611873} a^{8} + \frac{618288676513166086569710941688781878335261100}{2070113138601208678295936907053690503945744317} a^{7} - \frac{247203580845440049715640892271844843837474591}{2070113138601208678295936907053690503945744317} a^{6} - \frac{824526808985431618902428077795419406454293895}{2070113138601208678295936907053690503945744317} a^{5} - \frac{56818850538784157000148801004716509216245506}{230012570955689853143992989672632278216193813} a^{4} + \frac{748250827132988494229793453741192650416045394}{2070113138601208678295936907053690503945744317} a^{3} - \frac{687416137426693556095938237597643164579650521}{2070113138601208678295936907053690503945744317} a^{2} - \frac{100445112632963339064263381538512944882463431}{690037712867069559431978969017896834648581439} a - \frac{100767366496537500799257380228314441191340300}{690037712867069559431978969017896834648581439}$

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:  $13$
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:  \( 11134796611.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed