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Properties

Label 20.4.79692860778...7424.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{18}\cdot 241^{3}$
Root discriminant $27.87$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T747

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2861, -5187, -8023, 29238, -31388, 15848, -4391, 2299, -998, -1767, 2617, -1545, 961, -737, 357, -140, 90, -33, 8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 - 33*x^17 + 90*x^16 - 140*x^15 + 357*x^14 - 737*x^13 + 961*x^12 - 1545*x^11 + 2617*x^10 - 1767*x^9 - 998*x^8 + 2299*x^7 - 4391*x^6 + 15848*x^5 - 31388*x^4 + 29238*x^3 - 8023*x^2 - 5187*x + 2861)
 
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 - 33*x^17 + 90*x^16 - 140*x^15 + 357*x^14 - 737*x^13 + 961*x^12 - 1545*x^11 + 2617*x^10 - 1767*x^9 - 998*x^8 + 2299*x^7 - 4391*x^6 + 15848*x^5 - 31388*x^4 + 29238*x^3 - 8023*x^2 - 5187*x + 2861, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 8 x^{18} - 33 x^{17} + 90 x^{16} - 140 x^{15} + 357 x^{14} - 737 x^{13} + 961 x^{12} - 1545 x^{11} + 2617 x^{10} - 1767 x^{9} - 998 x^{8} + 2299 x^{7} - 4391 x^{6} + 15848 x^{5} - 31388 x^{4} + 29238 x^{3} - 8023 x^{2} - 5187 x + 2861 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(79692860778363999735970407424=2^{10}\cdot 11^{18}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.87$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2119492204796471948929306662991433542213} a^{19} - \frac{720199016757135618687719318460925368734}{2119492204796471948929306662991433542213} a^{18} - \frac{1010211072399847819245686536285457592280}{2119492204796471948929306662991433542213} a^{17} + \frac{1050263769607567680176315843274578596003}{2119492204796471948929306662991433542213} a^{16} - \frac{756888040144257550772346831353323608328}{2119492204796471948929306662991433542213} a^{15} + \frac{5063954172234008402472828090112762225}{2119492204796471948929306662991433542213} a^{14} + \frac{933402561305586491757766611514109132734}{2119492204796471948929306662991433542213} a^{13} - \frac{1055128673107260462090790662731550565001}{2119492204796471948929306662991433542213} a^{12} + \frac{953179435292416205019067789538326264514}{2119492204796471948929306662991433542213} a^{11} - \frac{401602755225943404663329117913978693150}{2119492204796471948929306662991433542213} a^{10} - \frac{963093040506970084475286107186221316965}{2119492204796471948929306662991433542213} a^{9} - \frac{294422217184058343798951187923304247055}{2119492204796471948929306662991433542213} a^{8} - \frac{283272446588849308697321368123184500142}{2119492204796471948929306662991433542213} a^{7} + \frac{1010472786811919657329733991359586298461}{2119492204796471948929306662991433542213} a^{6} - \frac{356651327969797143837569818624542126892}{2119492204796471948929306662991433542213} a^{5} + \frac{98733621435532750799265985704202384447}{2119492204796471948929306662991433542213} a^{4} - \frac{600600481753280538855000333712175182407}{2119492204796471948929306662991433542213} a^{3} + \frac{991233872051013951513058139082211630479}{2119492204796471948929306662991433542213} a^{2} - \frac{204611591150851755462432828727596095392}{2119492204796471948929306662991433542213} a - \frac{627954728094863544194133107327571912652}{2119492204796471948929306662991433542213}$

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:  $11$
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:  \( 2858560.65785 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T747:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n747 are not computed
Character table for t20n747 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R $20$ $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
2.10.10.4$x^{10} - 5 x^{8} + 14 x^{6} - 22 x^{4} + 17 x^{2} - 37$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
11Data not computed
241Data not computed