↑

Properties

Label 20.0.63834486136...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 3803^{4}$
Root discriminant $17.39$
Ramified primes $5, 3803$
Class number $1$
Class group Trivial
Galois group $C_4\times S_5$ (as 20T123)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 3, 6, 13, -2, -7, 11, 19, 0, -14, 17, 37, 10, 17, 10, 6, -1, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 2*x^18 - x^17 + 6*x^16 + 10*x^15 + 17*x^14 + 10*x^13 + 37*x^12 + 17*x^11 - 14*x^10 + 19*x^8 + 11*x^7 - 7*x^6 - 2*x^5 + 13*x^4 + 6*x^3 + 3*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^20 + 2*x^18 - x^17 + 6*x^16 + 10*x^15 + 17*x^14 + 10*x^13 + 37*x^12 + 17*x^11 - 14*x^10 + 19*x^8 + 11*x^7 - 7*x^6 - 2*x^5 + 13*x^4 + 6*x^3 + 3*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} + 2 x^{18} - x^{17} + 6 x^{16} + 10 x^{15} + 17 x^{14} + 10 x^{13} + 37 x^{12} + 17 x^{11} - 14 x^{10} + 19 x^{8} + 11 x^{7} - 7 x^{6} - 2 x^{5} + 13 x^{4} + 6 x^{3} + 3 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6383448613601104736328125=5^{15}\cdot 3803^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.39$
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:  $5, 3803$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{15} + \frac{2}{5} a^{14} + \frac{1}{5} a^{13} + \frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{61055} a^{17} + \frac{3273}{61055} a^{16} - \frac{25406}{61055} a^{15} + \frac{1309}{12211} a^{14} + \frac{6972}{61055} a^{13} + \frac{4982}{12211} a^{12} + \frac{3951}{61055} a^{11} - \frac{10053}{61055} a^{10} - \frac{9609}{61055} a^{9} + \frac{18791}{61055} a^{8} + \frac{5541}{61055} a^{7} + \frac{29556}{61055} a^{6} - \frac{2296}{61055} a^{5} - \frac{21348}{61055} a^{4} + \frac{19198}{61055} a^{3} - \frac{17572}{61055} a^{2} + \frac{22397}{61055} a + \frac{7842}{61055}$, $\frac{1}{61055} a^{18} - \frac{4466}{61055} a^{16} - \frac{8738}{61055} a^{15} - \frac{1786}{12211} a^{14} + \frac{27913}{61055} a^{13} - \frac{18054}{61055} a^{12} + \frac{1984}{61055} a^{11} - \frac{26996}{61055} a^{10} + \frac{25723}{61055} a^{9} - \frac{27228}{61055} a^{8} - \frac{21646}{61055} a^{7} - \frac{3542}{61055} a^{6} - \frac{3261}{12211} a^{5} - \frac{4562}{61055} a^{4} - \frac{2964}{12211} a^{3} + \frac{21743}{61055} a^{2} - \frac{7117}{61055} a + \frac{25078}{61055}$, $\frac{1}{305275} a^{19} - \frac{1}{305275} a^{18} - \frac{2}{305275} a^{17} + \frac{26466}{305275} a^{16} - \frac{4246}{61055} a^{15} - \frac{5651}{61055} a^{14} + \frac{12257}{305275} a^{13} + \frac{37123}{305275} a^{12} - \frac{97666}{305275} a^{11} - \frac{58347}{305275} a^{10} + \frac{96248}{305275} a^{9} + \frac{72302}{305275} a^{8} + \frac{135752}{305275} a^{7} + \frac{21999}{305275} a^{6} - \frac{41416}{305275} a^{5} + \frac{109024}{305275} a^{4} - \frac{58051}{305275} a^{3} - \frac{136703}{305275} a^{2} - \frac{81164}{305275} a - \frac{112804}{305275}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{10689}{61055} a^{19} - \frac{21378}{61055} a^{18} + \frac{32458}{61055} a^{17} - \frac{10689}{12211} a^{16} + \frac{21378}{12211} a^{15} - \frac{32067}{61055} a^{14} + \frac{32067}{61055} a^{13} - \frac{144772}{61055} a^{12} + \frac{363426}{61055} a^{11} - \frac{502383}{61055} a^{10} - \frac{117579}{61055} a^{9} + \frac{96201}{12211} a^{8} - \frac{1601}{61055} a^{7} - \frac{288603}{61055} a^{6} - \frac{21378}{12211} a^{5} + \frac{245847}{61055} a^{4} + \frac{21378}{12211} a^{3} - \frac{166781}{61055} a^{2} + \frac{42756}{61055} a + \frac{21378}{61055} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 33828.303351 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_5$ (as 20T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 480
The 28 conjugacy class representatives for $C_4\times S_5$
Character table for $C_4\times S_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.3.19015.1, 10.6.45196278125.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 24 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 $20$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
3803Data not computed