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Properties

Label 20.16.1256906660...8944.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{30}\cdot 11^{18}\cdot 1451^{2}$
Root discriminant $50.70$
Ramified primes $2, 11, 1451$
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![617, 7296, 6220, -101494, -228250, 26850, 265504, 155766, -33982, -124784, -61522, 32098, 33381, -1332, -7764, -584, 946, 70, -54, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 54*x^18 + 70*x^17 + 946*x^16 - 584*x^15 - 7764*x^14 - 1332*x^13 + 33381*x^12 + 32098*x^11 - 61522*x^10 - 124784*x^9 - 33982*x^8 + 155766*x^7 + 265504*x^6 + 26850*x^5 - 228250*x^4 - 101494*x^3 + 6220*x^2 + 7296*x + 617)
 
gp: K = bnfinit(x^20 - 2*x^19 - 54*x^18 + 70*x^17 + 946*x^16 - 584*x^15 - 7764*x^14 - 1332*x^13 + 33381*x^12 + 32098*x^11 - 61522*x^10 - 124784*x^9 - 33982*x^8 + 155766*x^7 + 265504*x^6 + 26850*x^5 - 228250*x^4 - 101494*x^3 + 6220*x^2 + 7296*x + 617, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 54 x^{18} + 70 x^{17} + 946 x^{16} - 584 x^{15} - 7764 x^{14} - 1332 x^{13} + 33381 x^{12} + 32098 x^{11} - 61522 x^{10} - 124784 x^{9} - 33982 x^{8} + 155766 x^{7} + 265504 x^{6} + 26850 x^{5} - 228250 x^{4} - 101494 x^{3} + 6220 x^{2} + 7296 x + 617 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12569066605710630176407970532818944=2^{30}\cdot 11^{18}\cdot 1451^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.70$
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:  $2, 11, 1451$
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}{196214458149955830345154303529017858641412793387783} a^{19} - \frac{86221045631750511637177907941482531512998374802604}{196214458149955830345154303529017858641412793387783} a^{18} - \frac{95231623927288794946408850883108870237463739614353}{196214458149955830345154303529017858641412793387783} a^{17} - \frac{36558776997674064047002926688063179212015002184460}{196214458149955830345154303529017858641412793387783} a^{16} - \frac{54378518551696399275045685807442711492580234596043}{196214458149955830345154303529017858641412793387783} a^{15} - \frac{88876776319590274219683860523293632015490238833679}{196214458149955830345154303529017858641412793387783} a^{14} + \frac{22726813153855458633646963496152661196395390919273}{196214458149955830345154303529017858641412793387783} a^{13} + \frac{93374911144752644743464465308998180145127347479715}{196214458149955830345154303529017858641412793387783} a^{12} + \frac{82160889381550694731011728916682535824433307772622}{196214458149955830345154303529017858641412793387783} a^{11} + \frac{50189853479791338573602178842143108773222964374069}{196214458149955830345154303529017858641412793387783} a^{10} + \frac{24695681907742240118983112352820369837708373258493}{196214458149955830345154303529017858641412793387783} a^{9} - \frac{1414649879194937894912857730562794397905307195321}{4563126933719903031282658221605066480032855660181} a^{8} - \frac{44418841366448832200602911024273065632512285514504}{196214458149955830345154303529017858641412793387783} a^{7} - \frac{15099520292727673829863597642724577862971757093241}{196214458149955830345154303529017858641412793387783} a^{6} + \frac{43798449746173319313794259533953637730329046501166}{196214458149955830345154303529017858641412793387783} a^{5} - \frac{29661731851847403697651986426825289195984898663669}{196214458149955830345154303529017858641412793387783} a^{4} - \frac{22252964670828163509212682978769512308304719616193}{196214458149955830345154303529017858641412793387783} a^{3} + \frac{314031310550417096722130443315751516095543052097}{4563126933719903031282658221605066480032855660181} a^{2} - \frac{55468705704404252869160317992636133786556136924339}{196214458149955830345154303529017858641412793387783} a + \frac{57939037040515841030438442724106000134874934636937}{196214458149955830345154303529017858641412793387783}$

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:  $17$
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:  \( 18451413891.6 \) (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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1451Data not computed