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Properties

Label 20.8.13617206346...4689.1
Degree $20$
Signature $[8, 6]$
Discriminant $11^{16}\cdot 23^{2}\cdot 571\cdot 9811$
Root discriminant $20.26$
Ramified primes $11, 23, 571, 9811$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T846

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 10, -30, 13, 52, 4, -45, -167, -27, 287, 87, -99, -50, -85, 54, 48, -34, 5, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 4*x^18 + 5*x^17 - 34*x^16 + 48*x^15 + 54*x^14 - 85*x^13 - 50*x^12 - 99*x^11 + 87*x^10 + 287*x^9 - 27*x^8 - 167*x^7 - 45*x^6 + 4*x^5 + 52*x^4 + 13*x^3 - 30*x^2 + 10*x - 1)
 
gp: K = bnfinit(x^20 - 4*x^19 + 4*x^18 + 5*x^17 - 34*x^16 + 48*x^15 + 54*x^14 - 85*x^13 - 50*x^12 - 99*x^11 + 87*x^10 + 287*x^9 - 27*x^8 - 167*x^7 - 45*x^6 + 4*x^5 + 52*x^4 + 13*x^3 - 30*x^2 + 10*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 4 x^{18} + 5 x^{17} - 34 x^{16} + 48 x^{15} + 54 x^{14} - 85 x^{13} - 50 x^{12} - 99 x^{11} + 87 x^{10} + 287 x^{9} - 27 x^{8} - 167 x^{7} - 45 x^{6} + 4 x^{5} + 52 x^{4} + 13 x^{3} - 30 x^{2} + 10 x - 1 \)

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:  \(136172063462016753296264689=11^{16}\cdot 23^{2}\cdot 571\cdot 9811\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.26$
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:  $11, 23, 571, 9811$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{1}{11} a^{14} + \frac{4}{11} a^{13} - \frac{1}{11} a^{12} - \frac{2}{11} a^{10} + \frac{3}{11} a^{9} - \frac{1}{11} a^{7} - \frac{2}{11} a^{6} - \frac{4}{11} a^{5} + \frac{1}{11} a^{4} - \frac{5}{11} a^{3} + \frac{3}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{3}{11} a^{14} + \frac{3}{11} a^{13} - \frac{1}{11} a^{12} - \frac{2}{11} a^{11} + \frac{1}{11} a^{10} + \frac{3}{11} a^{9} - \frac{1}{11} a^{8} - \frac{3}{11} a^{7} + \frac{5}{11} a^{6} - \frac{3}{11} a^{5} - \frac{4}{11} a^{4} - \frac{2}{11} a^{3} + \frac{5}{11} a^{2} + \frac{3}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{17} - \frac{5}{11} a^{14} - \frac{2}{11} a^{13} + \frac{1}{11} a^{12} + \frac{1}{11} a^{11} - \frac{2}{11} a^{10} + \frac{1}{11} a^{9} - \frac{3}{11} a^{8} - \frac{3}{11} a^{7} + \frac{3}{11} a^{6} - \frac{3}{11} a^{5} - \frac{5}{11} a^{4} - \frac{2}{11} a^{3} + \frac{5}{11} a^{2} - \frac{5}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{18} + \frac{4}{11} a^{14} - \frac{1}{11} a^{13} - \frac{4}{11} a^{12} - \frac{2}{11} a^{11} + \frac{2}{11} a^{10} + \frac{1}{11} a^{9} - \frac{3}{11} a^{8} - \frac{2}{11} a^{7} - \frac{2}{11} a^{6} - \frac{3}{11} a^{5} + \frac{3}{11} a^{4} + \frac{2}{11} a^{3} - \frac{1}{11} a^{2} - \frac{4}{11} a + \frac{5}{11}$, $\frac{1}{11125148895047} a^{19} - \frac{402168594434}{11125148895047} a^{18} + \frac{224783984857}{11125148895047} a^{17} - \frac{282032712605}{11125148895047} a^{16} + \frac{170926756131}{11125148895047} a^{15} + \frac{1793470859242}{11125148895047} a^{14} - \frac{1288839365849}{11125148895047} a^{13} + \frac{2089422663326}{11125148895047} a^{12} - \frac{4588960534203}{11125148895047} a^{11} + \frac{1008417956970}{11125148895047} a^{10} - \frac{4370290251362}{11125148895047} a^{9} + \frac{3852269748724}{11125148895047} a^{8} + \frac{790633860593}{11125148895047} a^{7} - \frac{3984058528658}{11125148895047} a^{6} - \frac{3563892558452}{11125148895047} a^{5} + \frac{2373694647}{1011377172277} a^{4} - \frac{843263005284}{11125148895047} a^{3} + \frac{5308499854065}{11125148895047} a^{2} - \frac{2656124737846}{11125148895047} a + \frac{164461591088}{11125148895047}$

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:  \( 239208.404354 \) (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

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

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$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$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
571Data not computed
9811Data not computed