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Properties

Label 20.4.69760000000...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{28}\cdot 5^{22}\cdot 109$
Root discriminant $19.60$
Ramified primes $2, 5, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T513

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 40, 160, 360, 600, 992, 1580, 1960, 1740, 1100, 500, 120, -100, -120, -50, -36, -20, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 20*x^16 - 36*x^15 - 50*x^14 - 120*x^13 - 100*x^12 + 120*x^11 + 500*x^10 + 1100*x^9 + 1740*x^8 + 1960*x^7 + 1580*x^6 + 992*x^5 + 600*x^4 + 360*x^3 + 160*x^2 + 40*x + 4)
 
gp: K = bnfinit(x^20 - 20*x^16 - 36*x^15 - 50*x^14 - 120*x^13 - 100*x^12 + 120*x^11 + 500*x^10 + 1100*x^9 + 1740*x^8 + 1960*x^7 + 1580*x^6 + 992*x^5 + 600*x^4 + 360*x^3 + 160*x^2 + 40*x + 4, 1)
 

Normalized defining polynomial

\( x^{20} - 20 x^{16} - 36 x^{15} - 50 x^{14} - 120 x^{13} - 100 x^{12} + 120 x^{11} + 500 x^{10} + 1100 x^{9} + 1740 x^{8} + 1960 x^{7} + 1580 x^{6} + 992 x^{5} + 600 x^{4} + 360 x^{3} + 160 x^{2} + 40 x + 4 \)

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:  \(69760000000000000000000000=2^{28}\cdot 5^{22}\cdot 109\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.60$
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:  $2, 5, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{76} a^{18} - \frac{9}{76} a^{17} - \frac{1}{76} a^{16} + \frac{3}{38} a^{15} + \frac{2}{19} a^{14} + \frac{2}{19} a^{13} - \frac{9}{38} a^{12} - \frac{3}{38} a^{11} - \frac{7}{38} a^{10} - \frac{11}{38} a^{9} - \frac{4}{19} a^{8} - \frac{15}{38} a^{7} - \frac{11}{38} a^{6} + \frac{5}{19} a^{5} + \frac{3}{19} a^{4} + \frac{1}{19} a^{3} + \frac{9}{19} a^{2} + \frac{3}{19} a - \frac{3}{19}$, $\frac{1}{19497577436097524} a^{19} - \frac{92541915593001}{19497577436097524} a^{18} + \frac{354356370549811}{19497577436097524} a^{17} - \frac{504798676956163}{19497577436097524} a^{16} + \frac{2144980847519037}{9748788718048762} a^{15} + \frac{134604207832317}{4874394359024381} a^{14} + \frac{1424551951278469}{9748788718048762} a^{13} + \frac{18654923882277}{168082564104289} a^{12} + \frac{766925732927084}{4874394359024381} a^{11} - \frac{2217572125205251}{9748788718048762} a^{10} - \frac{982928722588417}{4874394359024381} a^{9} + \frac{2007639364408892}{4874394359024381} a^{8} + \frac{2723398676339601}{9748788718048762} a^{7} + \frac{1817766174327723}{9748788718048762} a^{6} + \frac{1651940370291723}{4874394359024381} a^{5} - \frac{2263828832887925}{4874394359024381} a^{4} + \frac{1666742040058355}{4874394359024381} a^{3} - \frac{752229612195052}{4874394359024381} a^{2} + \frac{1451168869915196}{4874394359024381} a + \frac{138581130758238}{4874394359024381}$

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:  \( 108806.805365 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T513:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 128 conjugacy class representatives for t20n513 are not computed
Character table for t20n513 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.50000.1, 10.2.12500000000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
5Data not computed
109Data not computed