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Properties

Label 20.12.1077966233...2544.4
Degree $20$
Signature $[12, 4]$
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 20T340

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -97, -6, 870, -308, -2663, 3086, 1142, -4460, 2782, 424, -1289, 556, -67, 22, -21, -6, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 21*x^16 + 22*x^15 - 67*x^14 + 556*x^13 - 1289*x^12 + 424*x^11 + 2782*x^10 - 4460*x^9 + 1142*x^8 + 3086*x^7 - 2663*x^6 - 308*x^5 + 870*x^4 - 6*x^3 - 97*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 21*x^16 + 22*x^15 - 67*x^14 + 556*x^13 - 1289*x^12 + 424*x^11 + 2782*x^10 - 4460*x^9 + 1142*x^8 + 3086*x^7 - 2663*x^6 - 308*x^5 + 870*x^4 - 6*x^3 - 97*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 13 x^{18} - 6 x^{17} - 21 x^{16} + 22 x^{15} - 67 x^{14} + 556 x^{13} - 1289 x^{12} + 424 x^{11} + 2782 x^{10} - 4460 x^{9} + 1142 x^{8} + 3086 x^{7} - 2663 x^{6} - 308 x^{5} + 870 x^{4} - 6 x^{3} - 97 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{8} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{15} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{3}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{368} a^{18} + \frac{3}{92} a^{17} - \frac{11}{184} a^{16} - \frac{5}{184} a^{15} - \frac{15}{368} a^{14} - \frac{19}{184} a^{13} + \frac{35}{184} a^{12} - \frac{27}{184} a^{11} + \frac{41}{368} a^{10} + \frac{45}{92} a^{9} - \frac{37}{368} a^{8} + \frac{9}{184} a^{7} + \frac{81}{368} a^{6} - \frac{79}{184} a^{5} - \frac{39}{184} a^{4} - \frac{25}{184} a^{3} + \frac{11}{92} a^{2} + \frac{45}{92} a + \frac{11}{368}$, $\frac{1}{7236234951964908368} a^{19} + \frac{2596790426410229}{3618117475982454184} a^{18} - \frac{148141311996507271}{3618117475982454184} a^{17} + \frac{9162566472952477}{452264684497806773} a^{16} - \frac{53812613154255563}{7236234951964908368} a^{15} - \frac{262833372440157697}{3618117475982454184} a^{14} - \frac{180089998104577477}{3618117475982454184} a^{13} + \frac{2185050563618353}{157309455477498008} a^{12} + \frac{18069310264670443}{314618910954996016} a^{11} - \frac{627445232012331965}{3618117475982454184} a^{10} + \frac{58596048350670161}{314618910954996016} a^{9} - \frac{832826019439000765}{3618117475982454184} a^{8} - \frac{954961076959782791}{7236234951964908368} a^{7} + \frac{242642047370378029}{904529368995613546} a^{6} + \frac{1421375953891231491}{3618117475982454184} a^{5} - \frac{119942692066374295}{452264684497806773} a^{4} - \frac{836000612486747895}{1809058737991227092} a^{3} + \frac{231016686286895509}{3618117475982454184} a^{2} + \frac{2029770945591225995}{7236234951964908368} a - \frac{651892727766700013}{1809058737991227092}$

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:  $15$
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:  \( 5411981.38531 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T340:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 80 conjugacy class representatives for t20n340 are not computed
Character table for t20n340 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 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ 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}$
11Data not computed
$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$$0$Trivial$[\ ]$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.1.1$x^{2} - 43$$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}$