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Properties

Label 20.4.11475118187...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{13}\cdot 13^{6}\cdot 41^{7}$
Root discriminant $22.54$
Ramified primes $5, 13, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![941, -1183, -3631, 1100, 3713, -497, -917, 1589, 199, -957, 510, 598, -328, -60, 210, -37, -44, 26, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 26*x^17 - 44*x^16 - 37*x^15 + 210*x^14 - 60*x^13 - 328*x^12 + 598*x^11 + 510*x^10 - 957*x^9 + 199*x^8 + 1589*x^7 - 917*x^6 - 497*x^5 + 3713*x^4 + 1100*x^3 - 3631*x^2 - 1183*x + 941)
 
gp: K = bnfinit(x^20 - 4*x^19 + 26*x^17 - 44*x^16 - 37*x^15 + 210*x^14 - 60*x^13 - 328*x^12 + 598*x^11 + 510*x^10 - 957*x^9 + 199*x^8 + 1589*x^7 - 917*x^6 - 497*x^5 + 3713*x^4 + 1100*x^3 - 3631*x^2 - 1183*x + 941, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 26 x^{17} - 44 x^{16} - 37 x^{15} + 210 x^{14} - 60 x^{13} - 328 x^{12} + 598 x^{11} + 510 x^{10} - 957 x^{9} + 199 x^{8} + 1589 x^{7} - 917 x^{6} - 497 x^{5} + 3713 x^{4} + 1100 x^{3} - 3631 x^{2} - 1183 x + 941 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1147511818795502598876953125=5^{13}\cdot 13^{6}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.54$
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:  $5, 13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} + \frac{2}{13} a^{15} + \frac{5}{13} a^{14} - \frac{3}{13} a^{13} - \frac{6}{13} a^{12} - \frac{6}{13} a^{11} - \frac{5}{13} a^{10} + \frac{4}{13} a^{9} - \frac{6}{13} a^{8} - \frac{2}{13} a^{7} + \frac{1}{13} a^{6} + \frac{5}{13} a^{5} + \frac{6}{13} a^{4} - \frac{2}{13} a^{2} - \frac{4}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{17} + \frac{1}{13} a^{15} + \frac{6}{13} a^{12} - \frac{6}{13} a^{11} + \frac{1}{13} a^{10} - \frac{1}{13} a^{9} - \frac{3}{13} a^{8} + \frac{5}{13} a^{7} + \frac{3}{13} a^{6} - \frac{4}{13} a^{5} + \frac{1}{13} a^{4} - \frac{2}{13} a^{3} + \frac{3}{13}$, $\frac{1}{91} a^{18} - \frac{2}{91} a^{17} + \frac{1}{91} a^{16} + \frac{11}{91} a^{15} + \frac{3}{7} a^{14} + \frac{19}{91} a^{13} - \frac{5}{91} a^{12} + \frac{3}{7} a^{11} - \frac{16}{91} a^{10} + \frac{12}{91} a^{9} - \frac{15}{91} a^{8} + \frac{32}{91} a^{7} - \frac{36}{91} a^{6} - \frac{17}{91} a^{5} - \frac{4}{91} a^{4} - \frac{9}{91} a^{3} - \frac{23}{91} a + \frac{33}{91}$, $\frac{1}{28838595760939956908139066571} a^{19} + \frac{7090874030932863598623380}{28838595760939956908139066571} a^{18} + \frac{8522082111463592989990508}{4119799394419993844019866653} a^{17} - \frac{777476647547392930123723127}{28838595760939956908139066571} a^{16} - \frac{8736359462752587274457115961}{28838595760939956908139066571} a^{15} - \frac{2030324404380229491128724160}{4119799394419993844019866653} a^{14} + \frac{13175274940661791690921697160}{28838595760939956908139066571} a^{13} - \frac{911628972496099386793499457}{2218353520072304377549158967} a^{12} + \frac{158155573412870083606376695}{316907645724614911078451281} a^{11} - \frac{5590904008287065303316206715}{28838595760939956908139066571} a^{10} + \frac{2050729773575144511756305138}{28838595760939956908139066571} a^{9} - \frac{227378439644869565760206391}{4119799394419993844019866653} a^{8} - \frac{770732527865815215122343069}{2218353520072304377549158967} a^{7} - \frac{1169633247565981507948689727}{4119799394419993844019866653} a^{6} - \frac{1055232652114469185464729225}{28838595760939956908139066571} a^{5} + \frac{8137810773334457676445482802}{28838595760939956908139066571} a^{4} - \frac{3568874220326458287871531465}{28838595760939956908139066571} a^{3} + \frac{13360448530418417867993290980}{28838595760939956908139066571} a^{2} + \frac{13068419576921196531886241053}{28838595760939956908139066571} a - \frac{13800365830826305620719286885}{28838595760939956908139066571}$

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:  $11$
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:  \( 314730.217282 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.2665.1, 10.2.887778125.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$13$13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed