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Properties

Label 20.8.70638883973...9408.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{30}\cdot 3^{15}\cdot 71^{9}$
Root discriminant $43.90$
Ramified primes $2, 3, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T647

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, 278, 1380, -2680, -14092, 27772, 7998, -12022, -13642, -4440, 3570, 306, -257, -580, 60, -68, 114, -32, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 18*x^18 - 32*x^17 + 114*x^16 - 68*x^15 + 60*x^14 - 580*x^13 - 257*x^12 + 306*x^11 + 3570*x^10 - 4440*x^9 - 13642*x^8 - 12022*x^7 + 7998*x^6 + 27772*x^5 - 14092*x^4 - 2680*x^3 + 1380*x^2 + 278*x + 13)
 
gp: K = bnfinit(x^20 - 8*x^19 + 18*x^18 - 32*x^17 + 114*x^16 - 68*x^15 + 60*x^14 - 580*x^13 - 257*x^12 + 306*x^11 + 3570*x^10 - 4440*x^9 - 13642*x^8 - 12022*x^7 + 7998*x^6 + 27772*x^5 - 14092*x^4 - 2680*x^3 + 1380*x^2 + 278*x + 13, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 18 x^{18} - 32 x^{17} + 114 x^{16} - 68 x^{15} + 60 x^{14} - 580 x^{13} - 257 x^{12} + 306 x^{11} + 3570 x^{10} - 4440 x^{9} - 13642 x^{8} - 12022 x^{7} + 7998 x^{6} + 27772 x^{5} - 14092 x^{4} - 2680 x^{3} + 1380 x^{2} + 278 x + 13 \)

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:  \(706388839731582814105670315409408=2^{30}\cdot 3^{15}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.90$
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:  $2, 3, 71$
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}{108667129437596384441727616860824004538169167} a^{19} + \frac{19673043391591786179728304582281221885108981}{108667129437596384441727616860824004538169167} a^{18} + \frac{14322421598331395750585368361212030655410821}{108667129437596384441727616860824004538169167} a^{17} + \frac{28877019956543614430850399582576651441063409}{108667129437596384441727616860824004538169167} a^{16} + \frac{34403568789457000203927384721972939151734039}{108667129437596384441727616860824004538169167} a^{15} - \frac{50542726054369788033616620478283993139544516}{108667129437596384441727616860824004538169167} a^{14} + \frac{28074737655480872498425042807015599025569207}{108667129437596384441727616860824004538169167} a^{13} + \frac{12351246419892943197436674662166244835216489}{108667129437596384441727616860824004538169167} a^{12} + \frac{9524929537614511640595701995390004255062753}{108667129437596384441727616860824004538169167} a^{11} + \frac{28270174324758001019291679027428541479924754}{108667129437596384441727616860824004538169167} a^{10} + \frac{8622544575676484022504158016504213418329075}{108667129437596384441727616860824004538169167} a^{9} - \frac{50955057333343565053836807701955486479390797}{108667129437596384441727616860824004538169167} a^{8} - \frac{10745858393082755379274530010804349733805887}{108667129437596384441727616860824004538169167} a^{7} - \frac{332971971173941510589306612427153032693133}{108667129437596384441727616860824004538169167} a^{6} - \frac{16768103800478757907696738594302768294299397}{108667129437596384441727616860824004538169167} a^{5} - \frac{24462211440482035547635160790714545782526579}{108667129437596384441727616860824004538169167} a^{4} + \frac{21745624251291500498761001996336530971388511}{108667129437596384441727616860824004538169167} a^{3} + \frac{16449136321605779058849502969675918584287958}{108667129437596384441727616860824004538169167} a^{2} - \frac{20956690722738686181428229914215788419107846}{108667129437596384441727616860824004538169167} a - \frac{29574434893335480378643649042396961479577470}{108667129437596384441727616860824004538169167}$

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:  \( 637726535.732 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T647:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 51200
The 152 conjugacy class representatives for t20n647 are not computed
Character table for t20n647 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 10.10.6323239406592.1

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 R R $20$ $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
3Data not computed
$71$71.10.0.1$x^{10} - x + 22$$1$$10$$0$$C_{10}$$[\ ]^{10}$
71.10.9.7$x^{10} + 568$$10$$1$$9$$C_{10}$$[\ ]_{10}$