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Properties

Label 20.8.51714682171...4672.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{16}\cdot 13^{16}\cdot 17^{9}$
Root discriminant $48.49$
Ramified primes $2, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T803

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1733, 1248, -3297, 2250, -5705, -11602, 11828, 4264, -1065, 2940, -8641, 554, 5827, 336, -1444, -64, 189, 14, -17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 17*x^18 + 14*x^17 + 189*x^16 - 64*x^15 - 1444*x^14 + 336*x^13 + 5827*x^12 + 554*x^11 - 8641*x^10 + 2940*x^9 - 1065*x^8 + 4264*x^7 + 11828*x^6 - 11602*x^5 - 5705*x^4 + 2250*x^3 - 3297*x^2 + 1248*x + 1733)
 
gp: K = bnfinit(x^20 - 2*x^19 - 17*x^18 + 14*x^17 + 189*x^16 - 64*x^15 - 1444*x^14 + 336*x^13 + 5827*x^12 + 554*x^11 - 8641*x^10 + 2940*x^9 - 1065*x^8 + 4264*x^7 + 11828*x^6 - 11602*x^5 - 5705*x^4 + 2250*x^3 - 3297*x^2 + 1248*x + 1733, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 17 x^{18} + 14 x^{17} + 189 x^{16} - 64 x^{15} - 1444 x^{14} + 336 x^{13} + 5827 x^{12} + 554 x^{11} - 8641 x^{10} + 2940 x^{9} - 1065 x^{8} + 4264 x^{7} + 11828 x^{6} - 11602 x^{5} - 5705 x^{4} + 2250 x^{3} - 3297 x^{2} + 1248 x + 1733 \)

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:  \(5171468217146897008836483283484672=2^{16}\cdot 13^{16}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.49$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{20} a^{16} + \frac{1}{10} a^{15} + \frac{1}{5} a^{14} + \frac{1}{10} a^{13} + \frac{1}{5} a^{11} + \frac{3}{10} a^{9} - \frac{1}{4} a^{8} + \frac{2}{5} a^{7} + \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{10} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{3}{10} a - \frac{7}{20}$, $\frac{1}{20} a^{17} + \frac{1}{5} a^{14} - \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{10} + \frac{3}{20} a^{9} - \frac{1}{10} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{1}{10} a^{3} - \frac{1}{2} a^{2} + \frac{1}{20} a - \frac{3}{10}$, $\frac{1}{100} a^{18} - \frac{1}{100} a^{16} + \frac{3}{25} a^{15} + \frac{11}{50} a^{14} + \frac{11}{50} a^{13} + \frac{1}{50} a^{12} - \frac{2}{25} a^{11} - \frac{7}{100} a^{10} - \frac{19}{50} a^{9} + \frac{21}{100} a^{8} - \frac{13}{50} a^{7} - \frac{3}{50} a^{6} - \frac{7}{50} a^{5} + \frac{1}{25} a^{4} + \frac{8}{25} a^{3} - \frac{27}{100} a^{2} - \frac{21}{50} a - \frac{13}{100}$, $\frac{1}{45536053886790353524518951295398185284900} a^{19} + \frac{43348302671181154725123118482795719889}{45536053886790353524518951295398185284900} a^{18} - \frac{14329991300411283417765219993391949229}{11384013471697588381129737823849546321225} a^{17} + \frac{890978010638341044387672105833282781073}{45536053886790353524518951295398185284900} a^{16} - \frac{23707172305914605804918785691457733021}{98991421493022507661997720207387359315} a^{15} + \frac{620101729841770784446312222706347973627}{4553605388679035352451895129539818528490} a^{14} + \frac{5238358104756196727890545565345145302}{98991421493022507661997720207387359315} a^{13} - \frac{214112071072834528361441131548846685222}{2276802694339517676225947564769909264245} a^{12} + \frac{10909349284090314746440313428240999757701}{45536053886790353524518951295398185284900} a^{11} - \frac{7767854949240991369322838512947922822551}{45536053886790353524518951295398185284900} a^{10} + \frac{587932528471413362459131194008973641847}{22768026943395176762259475647699092642450} a^{9} - \frac{17119136283874071443794124662434650828977}{45536053886790353524518951295398185284900} a^{8} - \frac{1328162769338667145042459662101842074251}{4553605388679035352451895129539818528490} a^{7} + \frac{1897639757776343413900456875792886580793}{11384013471697588381129737823849546321225} a^{6} - \frac{84740869821580566758406086190346630203}{186623171667173580018520292194254857725} a^{5} - \frac{2483144486774779067709738855282281566373}{11384013471697588381129737823849546321225} a^{4} - \frac{14939289234793651144738351002490672047309}{45536053886790353524518951295398185284900} a^{3} - \frac{558497633362690161787659926576050457919}{9107210777358070704903790259079637056980} a^{2} + \frac{4675994564887614509737343615511914318767}{22768026943395176762259475647699092642450} a + \frac{12618020811198580955195602526019908644933}{45536053886790353524518951295398185284900}$

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:  \( 4764878974.43 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T803:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 126 conjugacy class representatives for t20n803 are not computed
Character table for t20n803 is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 5.5.10158928.1, 10.10.1341649635419392.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.12.11.6$x^{12} - 13312$$12$$1$$11$$C_{12}$$[\ ]_{12}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.5.2$x^{6} + 51$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$