↑

Properties

Label 20.16.5437529839...0000.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{40}\cdot 5^{12}\cdot 1193^{4}$
Root discriminant $43.33$
Ramified primes $2, 5, 1193$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T872

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 84, 290, -784, -4287, -3992, 4092, 6088, -5244, -10700, -122, 8312, 3440, -2612, -1776, 344, 361, -16, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 32*x^18 - 16*x^17 + 361*x^16 + 344*x^15 - 1776*x^14 - 2612*x^13 + 3440*x^12 + 8312*x^11 - 122*x^10 - 10700*x^9 - 5244*x^8 + 6088*x^7 + 4092*x^6 - 3992*x^5 - 4287*x^4 - 784*x^3 + 290*x^2 + 84*x + 2)
 
gp: K = bnfinit(x^20 - 32*x^18 - 16*x^17 + 361*x^16 + 344*x^15 - 1776*x^14 - 2612*x^13 + 3440*x^12 + 8312*x^11 - 122*x^10 - 10700*x^9 - 5244*x^8 + 6088*x^7 + 4092*x^6 - 3992*x^5 - 4287*x^4 - 784*x^3 + 290*x^2 + 84*x + 2, 1)
 

Normalized defining polynomial

\( x^{20} - 32 x^{18} - 16 x^{17} + 361 x^{16} + 344 x^{15} - 1776 x^{14} - 2612 x^{13} + 3440 x^{12} + 8312 x^{11} - 122 x^{10} - 10700 x^{9} - 5244 x^{8} + 6088 x^{7} + 4092 x^{6} - 3992 x^{5} - 4287 x^{4} - 784 x^{3} + 290 x^{2} + 84 x + 2 \)

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:  \(543752983985526931456000000000000=2^{40}\cdot 5^{12}\cdot 1193^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.33$
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:  $2, 5, 1193$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{11} + \frac{1}{5} a^{8} - \frac{1}{5} a^{4} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{18} + \frac{1}{25} a^{16} - \frac{1}{25} a^{15} - \frac{1}{25} a^{14} + \frac{1}{25} a^{13} + \frac{1}{25} a^{12} + \frac{1}{25} a^{11} + \frac{8}{25} a^{10} + \frac{12}{25} a^{8} - \frac{2}{5} a^{7} + \frac{7}{25} a^{6} + \frac{8}{25} a^{5} - \frac{2}{25} a^{4} + \frac{12}{25} a^{3} + \frac{12}{25} a^{2} + \frac{2}{25} a - \frac{9}{25}$, $\frac{1}{236314091277148461575} a^{19} - \frac{642634546689148467}{236314091277148461575} a^{18} + \frac{20990727246050816016}{236314091277148461575} a^{17} + \frac{16710971025369967712}{236314091277148461575} a^{16} - \frac{15972323532910805069}{236314091277148461575} a^{15} + \frac{4495191688073618318}{236314091277148461575} a^{14} + \frac{5553326054135458189}{236314091277148461575} a^{13} - \frac{837158228556826807}{10274525707702107025} a^{12} + \frac{55588258202814324056}{236314091277148461575} a^{11} - \frac{110756551579244430961}{236314091277148461575} a^{10} - \frac{60011213325859135738}{236314091277148461575} a^{9} + \frac{69341819162042449471}{236314091277148461575} a^{8} - \frac{16568793891635725918}{236314091277148461575} a^{7} - \frac{15101248437182619696}{236314091277148461575} a^{6} + \frac{80235399763850228992}{236314091277148461575} a^{5} - \frac{12551495508974701089}{236314091277148461575} a^{4} - \frac{4065058827798491257}{236314091277148461575} a^{3} + \frac{11766829469311351853}{236314091277148461575} a^{2} - \frac{100466057259055596938}{236314091277148461575} a + \frac{59207463382761152713}{236314091277148461575}$

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:  \( 8111058170.51 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 204800
The 116 conjugacy class representatives for t20n872 are not computed
Character table for t20n872 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.728703488000000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
1193Data not computed