↑

Properties

Label 20.10.1825672741...6811.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,11^{16}\cdot 331^{5}$
Root discriminant $29.04$
Ramified primes $11, 331$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times S_4$ (as 20T34)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 18, -49, 74, 234, -473, -334, 350, 348, 174, -271, -79, 94, -81, -3, 12, 2, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 5*x^18 + 2*x^17 + 12*x^16 - 3*x^15 - 81*x^14 + 94*x^13 - 79*x^12 - 271*x^11 + 174*x^10 + 348*x^9 + 350*x^8 - 334*x^7 - 473*x^6 + 234*x^5 + 74*x^4 - 49*x^3 + 18*x^2 + 3*x - 1)
 
gp: K = bnfinit(x^20 - x^19 + 5*x^18 + 2*x^17 + 12*x^16 - 3*x^15 - 81*x^14 + 94*x^13 - 79*x^12 - 271*x^11 + 174*x^10 + 348*x^9 + 350*x^8 - 334*x^7 - 473*x^6 + 234*x^5 + 74*x^4 - 49*x^3 + 18*x^2 + 3*x - 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 5 x^{18} + 2 x^{17} + 12 x^{16} - 3 x^{15} - 81 x^{14} + 94 x^{13} - 79 x^{12} - 271 x^{11} + 174 x^{10} + 348 x^{9} + 350 x^{8} - 334 x^{7} - 473 x^{6} + 234 x^{5} + 74 x^{4} - 49 x^{3} + 18 x^{2} + 3 x - 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-182567274194490055859030886811=-\,11^{16}\cdot 331^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.04$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{20874808829546289995981} a^{19} - \frac{569535176549815533991}{20874808829546289995981} a^{18} - \frac{5140524185091686924922}{20874808829546289995981} a^{17} + \frac{9747547766803746797298}{20874808829546289995981} a^{16} + \frac{7911204148518638343248}{20874808829546289995981} a^{15} + \frac{5070409723275315187757}{20874808829546289995981} a^{14} - \frac{3913075469530983980394}{20874808829546289995981} a^{13} - \frac{4426926777651939971571}{20874808829546289995981} a^{12} - \frac{4220843419743088189431}{20874808829546289995981} a^{11} + \frac{5475242833698289929525}{20874808829546289995981} a^{10} + \frac{8416498641045256808085}{20874808829546289995981} a^{9} - \frac{540563298864852178977}{20874808829546289995981} a^{8} - \frac{8406363485823949494327}{20874808829546289995981} a^{7} + \frac{1353448626534231713957}{20874808829546289995981} a^{6} - \frac{4421629357016738729824}{20874808829546289995981} a^{5} - \frac{4313878914423339127072}{20874808829546289995981} a^{4} + \frac{7403405894661065302451}{20874808829546289995981} a^{3} - \frac{7360647048760299975685}{20874808829546289995981} a^{2} + \frac{8021017324420205738628}{20874808829546289995981} a - \frac{2134791760788774560351}{20874808829546289995981}$

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:  $14$
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:  \( 18726448.7284 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times S_4$ (as 20T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 25 conjugacy class representatives for $C_5\times S_4$
Character table for $C_5\times S_4$ is not computed

Intermediate fields

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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: data not computed
Degree 40 sibling: 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 $20$ $20$ $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ $15{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ $15{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ $20$ $15{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ $20$ $20$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ $15{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
11Data not computed
331Data not computed