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Properties

Label 20.8.10779662339...2544.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 11^{18}\cdot 43^{2}$
Root discriminant $25.21$
Ramified primes $2, 11, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T262

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43, 516, -1904, 2346, -408, -1534, 1008, 1158, -1029, -682, 100, 924, -431, -130, 178, -72, -9, 10, 8, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 8*x^18 + 10*x^17 - 9*x^16 - 72*x^15 + 178*x^14 - 130*x^13 - 431*x^12 + 924*x^11 + 100*x^10 - 682*x^9 - 1029*x^8 + 1158*x^7 + 1008*x^6 - 1534*x^5 - 408*x^4 + 2346*x^3 - 1904*x^2 + 516*x - 43)
 
gp: K = bnfinit(x^20 - 6*x^19 + 8*x^18 + 10*x^17 - 9*x^16 - 72*x^15 + 178*x^14 - 130*x^13 - 431*x^12 + 924*x^11 + 100*x^10 - 682*x^9 - 1029*x^8 + 1158*x^7 + 1008*x^6 - 1534*x^5 - 408*x^4 + 2346*x^3 - 1904*x^2 + 516*x - 43, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 8 x^{18} + 10 x^{17} - 9 x^{16} - 72 x^{15} + 178 x^{14} - 130 x^{13} - 431 x^{12} + 924 x^{11} + 100 x^{10} - 682 x^{9} - 1029 x^{8} + 1158 x^{7} + 1008 x^{6} - 1534 x^{5} - 408 x^{4} + 2346 x^{3} - 1904 x^{2} + 516 x - 43 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10779662339431083287111532544=2^{20}\cdot 11^{18}\cdot 43^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.21$
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:  $2, 11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{110} a^{15} - \frac{12}{55} a^{14} + \frac{2}{11} a^{13} - \frac{21}{110} a^{12} + \frac{21}{110} a^{10} + \frac{21}{55} a^{9} + \frac{7}{110} a^{8} - \frac{19}{110} a^{7} - \frac{49}{110} a^{6} + \frac{9}{22} a^{5} - \frac{1}{5} a^{4} - \frac{5}{22} a^{3} + \frac{1}{110} a^{2} - \frac{26}{55} a - \frac{6}{55}$, $\frac{1}{110} a^{16} - \frac{3}{55} a^{14} + \frac{19}{110} a^{13} - \frac{9}{110} a^{12} + \frac{21}{110} a^{11} - \frac{2}{55} a^{10} + \frac{5}{22} a^{9} + \frac{39}{110} a^{8} + \frac{9}{22} a^{7} - \frac{31}{110} a^{6} - \frac{21}{55} a^{5} + \frac{26}{55} a^{4} - \frac{49}{110} a^{3} + \frac{27}{110} a^{2} - \frac{5}{11} a - \frac{13}{110}$, $\frac{1}{110} a^{17} - \frac{3}{22} a^{14} + \frac{1}{110} a^{13} + \frac{1}{22} a^{12} - \frac{2}{55} a^{11} - \frac{7}{55} a^{10} - \frac{39}{110} a^{9} + \frac{16}{55} a^{8} - \frac{7}{22} a^{7} + \frac{49}{110} a^{6} - \frac{4}{55} a^{5} - \frac{8}{55} a^{4} - \frac{13}{110} a^{3} - \frac{2}{5} a^{2} + \frac{1}{22} a - \frac{17}{110}$, $\frac{1}{110} a^{18} + \frac{13}{55} a^{14} - \frac{5}{22} a^{13} + \frac{1}{10} a^{12} - \frac{7}{55} a^{11} + \frac{1}{110} a^{10} + \frac{1}{55} a^{9} + \frac{3}{22} a^{8} - \frac{8}{55} a^{7} + \frac{27}{110} a^{6} - \frac{1}{110} a^{5} - \frac{13}{110} a^{4} + \frac{21}{110} a^{3} - \frac{7}{22} a^{2} - \frac{27}{110} a - \frac{3}{22}$, $\frac{1}{219364338066481497670} a^{19} + \frac{29402839976618828}{9971106275749158985} a^{18} - \frac{75269622554929861}{43872867613296299534} a^{17} + \frac{366173980992004409}{109682169033240748835} a^{16} + \frac{60318848864497}{43872867613296299534} a^{15} + \frac{17386084593075841047}{219364338066481497670} a^{14} - \frac{21475548461666031387}{219364338066481497670} a^{13} - \frac{271668314949746042}{109682169033240748835} a^{12} - \frac{5992158247983549913}{43872867613296299534} a^{11} - \frac{3609633408326892170}{21936433806648149767} a^{10} + \frac{3841545439769879977}{21936433806648149767} a^{9} + \frac{18155459908482506837}{109682169033240748835} a^{8} + \frac{17084369112580823935}{43872867613296299534} a^{7} + \frac{102592382398643887987}{219364338066481497670} a^{6} - \frac{11276265794774634119}{43872867613296299534} a^{5} - \frac{80091492637791709189}{219364338066481497670} a^{4} - \frac{30532321312195858073}{109682169033240748835} a^{3} + \frac{65867176608045983403}{219364338066481497670} a^{2} - \frac{1456158794795323167}{43872867613296299534} a - \frac{42833668836394222211}{109682169033240748835}$

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:  \( 2808419.53416 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T262:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 40 conjugacy class representatives for t20n262
Character table for t20n262 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 32 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.1$x^{2} - 43$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$