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Properties

Label 20.6.78574805586...9483.2
Degree $20$
Signature $[6, 7]$
Discriminant $-\,13^{10}\cdot 97^{2}\cdot 347^{7}$
Root discriminant $44.13$
Ramified primes $13, 97, 347$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T796

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3073379, 5057872, -176876, -3458400, 720336, 1100891, 1152018, -2570439, 1056435, 573446, -645738, 127471, 80856, -47747, 5893, 2647, -1287, 218, 8, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^19 + 8*x^18 + 218*x^17 - 1287*x^16 + 2647*x^15 + 5893*x^14 - 47747*x^13 + 80856*x^12 + 127471*x^11 - 645738*x^10 + 573446*x^9 + 1056435*x^8 - 2570439*x^7 + 1152018*x^6 + 1100891*x^5 + 720336*x^4 - 3458400*x^3 - 176876*x^2 + 5057872*x - 3073379)
 
gp: K = bnfinit(x^20 - 9*x^19 + 8*x^18 + 218*x^17 - 1287*x^16 + 2647*x^15 + 5893*x^14 - 47747*x^13 + 80856*x^12 + 127471*x^11 - 645738*x^10 + 573446*x^9 + 1056435*x^8 - 2570439*x^7 + 1152018*x^6 + 1100891*x^5 + 720336*x^4 - 3458400*x^3 - 176876*x^2 + 5057872*x - 3073379, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{19} + 8 x^{18} + 218 x^{17} - 1287 x^{16} + 2647 x^{15} + 5893 x^{14} - 47747 x^{13} + 80856 x^{12} + 127471 x^{11} - 645738 x^{10} + 573446 x^{9} + 1056435 x^{8} - 2570439 x^{7} + 1152018 x^{6} + 1100891 x^{5} + 720336 x^{4} - 3458400 x^{3} - 176876 x^{2} + 5057872 x - 3073379 \)

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:  \(-785748055861331850537130300409483=-\,13^{10}\cdot 97^{2}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.13$
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:  $13, 97, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{292251780310492828878042174892612275688273985883697409231589503} a^{19} + \frac{100306521317593036924332447036288409439018221118384862082716608}{292251780310492828878042174892612275688273985883697409231589503} a^{18} - \frac{3907155298791224982848609513593644050667218938661411252735575}{292251780310492828878042174892612275688273985883697409231589503} a^{17} - \frac{2068679637033836456320439264319792420477935196141317348781754}{292251780310492828878042174892612275688273985883697409231589503} a^{16} - \frac{137641036071095960728086259300793238337555590779263887804728717}{292251780310492828878042174892612275688273985883697409231589503} a^{15} - \frac{144554065184724939193215230688651629575347551841992979791993186}{292251780310492828878042174892612275688273985883697409231589503} a^{14} - \frac{121416796807611044506243629990038890127770728451153713462222620}{292251780310492828878042174892612275688273985883697409231589503} a^{13} + \frac{80477526768315745784807512648195450202480311889084536652639689}{292251780310492828878042174892612275688273985883697409231589503} a^{12} + \frac{83476294352973418974033276811491763085354591933146318895336806}{292251780310492828878042174892612275688273985883697409231589503} a^{11} - \frac{113311552107901958448087594673254481820959422503729355786289162}{292251780310492828878042174892612275688273985883697409231589503} a^{10} - \frac{94440283123892836673389407441901174188211854582205271436337017}{292251780310492828878042174892612275688273985883697409231589503} a^{9} + \frac{26712749712863779718043416923794654019777077912371534206665555}{292251780310492828878042174892612275688273985883697409231589503} a^{8} + \frac{144769235758640664656429991759953297228673543543864851598188174}{292251780310492828878042174892612275688273985883697409231589503} a^{7} + \frac{105719393681016860101929988811503349637254041646439212674062070}{292251780310492828878042174892612275688273985883697409231589503} a^{6} + \frac{119428130467207780894630197336984079721213640223726185023028128}{292251780310492828878042174892612275688273985883697409231589503} a^{5} + \frac{20279952856684823229165412445968068315154044704729153125947430}{292251780310492828878042174892612275688273985883697409231589503} a^{4} - \frac{43538953486672504632822700425478126208418675345459080718233209}{292251780310492828878042174892612275688273985883697409231589503} a^{3} - \frac{116971139165146875349501934232266478974409192103298726618355}{3699389624183453530101799682184965515041442859287308977615057} a^{2} - \frac{89705507843659068692481350337723154441248855830266397212724057}{292251780310492828878042174892612275688273985883697409231589503} a - \frac{118341297944381807940552040711853345168783629988508337221170159}{292251780310492828878042174892612275688273985883697409231589503}$

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:  \( 549169238.072 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n796 are not computed
Character table for t20n796 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 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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{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$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
347Data not computed