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Properties

Label 20.8.57722337794...4089.6
Degree $20$
Signature $[8, 6]$
Discriminant $11^{16}\cdot 23^{4}\cdot 67^{2}$
Root discriminant $19.41$
Ramified primes $11, 23, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T341

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -19, 129, -482, 1054, 355, -2361, 433, 1507, -182, -664, 186, 242, -309, 75, 74, -51, 14, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 2*x^18 + 14*x^17 - 51*x^16 + 74*x^15 + 75*x^14 - 309*x^13 + 242*x^12 + 186*x^11 - 664*x^10 - 182*x^9 + 1507*x^8 + 433*x^7 - 2361*x^6 + 355*x^5 + 1054*x^4 - 482*x^3 + 129*x^2 - 19*x + 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 2*x^18 + 14*x^17 - 51*x^16 + 74*x^15 + 75*x^14 - 309*x^13 + 242*x^12 + 186*x^11 - 664*x^10 - 182*x^9 + 1507*x^8 + 433*x^7 - 2361*x^6 + 355*x^5 + 1054*x^4 - 482*x^3 + 129*x^2 - 19*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 2 x^{18} + 14 x^{17} - 51 x^{16} + 74 x^{15} + 75 x^{14} - 309 x^{13} + 242 x^{12} + 186 x^{11} - 664 x^{10} - 182 x^{9} + 1507 x^{8} + 433 x^{7} - 2361 x^{6} + 355 x^{5} + 1054 x^{4} - 482 x^{3} + 129 x^{2} - 19 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57722337794481266110634089=11^{16}\cdot 23^{4}\cdot 67^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.41$
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:  $11, 23, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{989} a^{18} + \frac{160}{989} a^{17} - \frac{180}{989} a^{16} - \frac{370}{989} a^{15} + \frac{446}{989} a^{14} - \frac{93}{989} a^{13} + \frac{370}{989} a^{12} - \frac{377}{989} a^{11} - \frac{143}{989} a^{10} + \frac{356}{989} a^{9} - \frac{265}{989} a^{8} + \frac{21}{989} a^{7} - \frac{284}{989} a^{6} + \frac{307}{989} a^{5} - \frac{169}{989} a^{4} - \frac{434}{989} a^{3} + \frac{80}{989} a^{2} + \frac{463}{989} a - \frac{359}{989}$, $\frac{1}{11448412577334272415041} a^{19} - \frac{2844608837228344278}{11448412577334272415041} a^{18} + \frac{3320235650609497536248}{11448412577334272415041} a^{17} - \frac{3984662002409722783247}{11448412577334272415041} a^{16} - \frac{5162031087498446511079}{11448412577334272415041} a^{15} + \frac{563797237651814323011}{11448412577334272415041} a^{14} + \frac{4649574803283532956988}{11448412577334272415041} a^{13} - \frac{1326110105260142268265}{11448412577334272415041} a^{12} - \frac{3409633516646457728065}{11448412577334272415041} a^{11} - \frac{1035529712914640427741}{11448412577334272415041} a^{10} - \frac{5203471019948923358338}{11448412577334272415041} a^{9} - \frac{4860978928720413833152}{11448412577334272415041} a^{8} + \frac{1440773011714928236728}{11448412577334272415041} a^{7} - \frac{5005609468261713460139}{11448412577334272415041} a^{6} - \frac{456443706947690538760}{11448412577334272415041} a^{5} - \frac{35277341284628741547}{11448412577334272415041} a^{4} - \frac{2107272604450414153377}{11448412577334272415041} a^{3} + \frac{2209496072377979050793}{11448412577334272415041} a^{2} - \frac{3158819599193425851458}{11448412577334272415041} a + \frac{32592023107914579638}{266242152961262149187}$

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:  \( 172041.698841 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T341:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 224 conjugacy class representatives for t20n341 are not computed
Character table for t20n341 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.330327035621.4, 10.8.7597521819283.4, 10.4.4930254263.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$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}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$