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Properties

Label 20.8.55156276191...0081.1
Degree $20$
Signature $[8, 6]$
Discriminant $11^{16}\cdot 331^{4}$
Root discriminant $21.73$
Ramified primes $11, 331$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T254

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 10, -61, -4, 322, -470, 440, -1008, 2133, -2895, 2735, -1796, 632, 195, -454, 320, -111, 1, 17, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 17*x^18 + x^17 - 111*x^16 + 320*x^15 - 454*x^14 + 195*x^13 + 632*x^12 - 1796*x^11 + 2735*x^10 - 2895*x^9 + 2133*x^8 - 1008*x^7 + 440*x^6 - 470*x^5 + 322*x^4 - 4*x^3 - 61*x^2 + 10*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 17*x^18 + x^17 - 111*x^16 + 320*x^15 - 454*x^14 + 195*x^13 + 632*x^12 - 1796*x^11 + 2735*x^10 - 2895*x^9 + 2133*x^8 - 1008*x^7 + 440*x^6 - 470*x^5 + 322*x^4 - 4*x^3 - 61*x^2 + 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 17 x^{18} + x^{17} - 111 x^{16} + 320 x^{15} - 454 x^{14} + 195 x^{13} + 632 x^{12} - 1796 x^{11} + 2735 x^{10} - 2895 x^{9} + 2133 x^{8} - 1008 x^{7} + 440 x^{6} - 470 x^{5} + 322 x^{4} - 4 x^{3} - 61 x^{2} + 10 x + 1 \)

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:  \(551562761916888386281060081=11^{16}\cdot 331^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.73$
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:  $11, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2569028650946696750041} a^{19} - \frac{331145655549638676231}{2569028650946696750041} a^{18} + \frac{806260092761144288603}{2569028650946696750041} a^{17} + \frac{397489429265494375186}{2569028650946696750041} a^{16} - \frac{645736714796202458205}{2569028650946696750041} a^{15} - \frac{234385093982827691314}{2569028650946696750041} a^{14} - \frac{724140785890466727922}{2569028650946696750041} a^{13} + \frac{146930285667210132880}{2569028650946696750041} a^{12} - \frac{1247107265211032390642}{2569028650946696750041} a^{11} + \frac{127521088041784358073}{2569028650946696750041} a^{10} + \frac{1097869636241014197706}{2569028650946696750041} a^{9} + \frac{563478868908600336733}{2569028650946696750041} a^{8} + \frac{666022063753520199562}{2569028650946696750041} a^{7} + \frac{244787654378157086440}{2569028650946696750041} a^{6} - \frac{1095808881534293889616}{2569028650946696750041} a^{5} + \frac{375131915820712187398}{2569028650946696750041} a^{4} + \frac{936815999470464729626}{2569028650946696750041} a^{3} - \frac{910410493100127396791}{2569028650946696750041} a^{2} - \frac{574495928195519227477}{2569028650946696750041} a - \frac{346245887644173131685}{2569028650946696750041}$

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:  \( 507963.53503 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T254:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 28 conjugacy class representatives for t20n254
Character table for t20n254 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.8.70952789611.1

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
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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
331Data not computed