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Properties

Label 20.4.15872710287...4704.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 11^{16}\cdot 241^{4}$
Root discriminant $28.84$
Ramified primes $2, 11, 241$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1231, -3720, 4456, -5626, 1937, 2276, -1067, 475, 6148, -1157, -2585, 1251, 284, -579, 237, 80, -41, 27, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 4*x^18 + 27*x^17 - 41*x^16 + 80*x^15 + 237*x^14 - 579*x^13 + 284*x^12 + 1251*x^11 - 2585*x^10 - 1157*x^9 + 6148*x^8 + 475*x^7 - 1067*x^6 + 2276*x^5 + 1937*x^4 - 5626*x^3 + 4456*x^2 - 3720*x + 1231)
 
gp: K = bnfinit(x^20 - x^19 + 4*x^18 + 27*x^17 - 41*x^16 + 80*x^15 + 237*x^14 - 579*x^13 + 284*x^12 + 1251*x^11 - 2585*x^10 - 1157*x^9 + 6148*x^8 + 475*x^7 - 1067*x^6 + 2276*x^5 + 1937*x^4 - 5626*x^3 + 4456*x^2 - 3720*x + 1231, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 4 x^{18} + 27 x^{17} - 41 x^{16} + 80 x^{15} + 237 x^{14} - 579 x^{13} + 284 x^{12} + 1251 x^{11} - 2585 x^{10} - 1157 x^{9} + 6148 x^{8} + 475 x^{7} - 1067 x^{6} + 2276 x^{5} + 1937 x^{4} - 5626 x^{3} + 4456 x^{2} - 3720 x + 1231 \)

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:  \(158727102872609288730321224704=2^{10}\cdot 11^{16}\cdot 241^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.84$
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}{2831658504045616451416930484761052624269967} a^{19} + \frac{908852934589895475426357027225574111512468}{2831658504045616451416930484761052624269967} a^{18} + \frac{1300073217821535194881820267964771962385569}{2831658504045616451416930484761052624269967} a^{17} + \frac{33179625863200887531002725784268022903350}{2831658504045616451416930484761052624269967} a^{16} + \frac{37444425795888643788105418169832327848230}{2831658504045616451416930484761052624269967} a^{15} - \frac{157152542040409969031608219998402048319135}{2831658504045616451416930484761052624269967} a^{14} + \frac{153994352535061318158115305161578012362372}{2831658504045616451416930484761052624269967} a^{13} + \frac{1306981638057608687747058752053926120471384}{2831658504045616451416930484761052624269967} a^{12} - \frac{684792281431119152414416465282380272348991}{2831658504045616451416930484761052624269967} a^{11} + \frac{66218443784423602547683482827642082215310}{2831658504045616451416930484761052624269967} a^{10} + \frac{503798529700496903405530355256696870713224}{2831658504045616451416930484761052624269967} a^{9} - \frac{682462192202404873378020175929552006678982}{2831658504045616451416930484761052624269967} a^{8} + \frac{756137708429860840407074137176722835443232}{2831658504045616451416930484761052624269967} a^{7} - \frac{1383839036082179678000149807192691083890942}{2831658504045616451416930484761052624269967} a^{6} - \frac{248046138844253465189756187724425027066995}{2831658504045616451416930484761052624269967} a^{5} + \frac{256357168867777620037037550010124486888838}{2831658504045616451416930484761052624269967} a^{4} + \frac{560147117186552608065915153648171222488541}{2831658504045616451416930484761052624269967} a^{3} + \frac{190821198768276129502474963096097606660588}{2831658504045616451416930484761052624269967} a^{2} + \frac{701521878953264102684447803814111294448902}{2831658504045616451416930484761052624269967} a + \frac{982288536466875172081727450444879274021369}{2831658504045616451416930484761052624269967}$

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:  \( 3364853.93864 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

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 t20n751 are not computed
Character table for t20n751 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$2$2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$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}$
241Data not computed