↑

Properties

Label 20.2.65861868411...0144.2
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{10}\cdot 11^{16}\cdot 241^{3}$
Root discriminant $21.92$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43, -815, -1008, -1141, -948, 35, -295, -307, -145, 20, 103, -43, 52, -32, -1, -11, 8, -5, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^17 + 8*x^16 - 11*x^15 - x^14 - 32*x^13 + 52*x^12 - 43*x^11 + 103*x^10 + 20*x^9 - 145*x^8 - 307*x^7 - 295*x^6 + 35*x^5 - 948*x^4 - 1141*x^3 - 1008*x^2 - 815*x + 43)
 
gp: K = bnfinit(x^20 - 5*x^17 + 8*x^16 - 11*x^15 - x^14 - 32*x^13 + 52*x^12 - 43*x^11 + 103*x^10 + 20*x^9 - 145*x^8 - 307*x^7 - 295*x^6 + 35*x^5 - 948*x^4 - 1141*x^3 - 1008*x^2 - 815*x + 43, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{17} + 8 x^{16} - 11 x^{15} - x^{14} - 32 x^{13} + 52 x^{12} - 43 x^{11} + 103 x^{10} + 20 x^{9} - 145 x^{8} - 307 x^{7} - 295 x^{6} + 35 x^{5} - 948 x^{4} - 1141 x^{3} - 1008 x^{2} - 815 x + 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-658618684118710741619590144=-\,2^{10}\cdot 11^{16}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.92$
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:  $2, 11, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{771405430862872360796739029039377} a^{19} + \frac{139238383009766711292036312158055}{771405430862872360796739029039377} a^{18} - \frac{193956381158513150102224559925727}{771405430862872360796739029039377} a^{17} + \frac{43602331456571735261552456101784}{771405430862872360796739029039377} a^{16} - \frac{83223266255162778672083594863548}{771405430862872360796739029039377} a^{15} + \frac{381129802054017525253400094061703}{771405430862872360796739029039377} a^{14} - \frac{356928305106765706840991014195677}{771405430862872360796739029039377} a^{13} - \frac{316898063814711993787895028382769}{771405430862872360796739029039377} a^{12} + \frac{114519317084239073443737699431523}{771405430862872360796739029039377} a^{11} - \frac{113924972288778961378406608160889}{771405430862872360796739029039377} a^{10} - \frac{216077267574361650863190259789443}{771405430862872360796739029039377} a^{9} - \frac{87282232064172476587932328994584}{771405430862872360796739029039377} a^{8} - \frac{248514575213564300541654475019473}{771405430862872360796739029039377} a^{7} - \frac{165258796961849595133374415247737}{771405430862872360796739029039377} a^{6} - \frac{157929570914456249990309780761119}{771405430862872360796739029039377} a^{5} + \frac{144145822759702813689742899819840}{771405430862872360796739029039377} a^{4} - \frac{300373368538586049328671016938825}{771405430862872360796739029039377} a^{3} + \frac{112816020859169306318925057428483}{771405430862872360796739029039377} a^{2} - \frac{283697371122039267330382127123145}{771405430862872360796739029039377} a + \frac{84908238034490125823516270491812}{771405430862872360796739029039377}$

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:  $10$
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:  \( 156416.723214 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T846:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 163840
The 649 conjugacy class representatives for t20n846 are not computed
Character table for t20n846 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.51660490321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.10.10.14$x^{10} + 5 x^{8} - 50 x^{6} - 58 x^{4} + 49 x^{2} + 21$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
241Data not computed