↑

Properties

Label 20.0.67729133686...9952.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 11^{16}\cdot 14057$
Root discriminant $21.96$
Ramified primes $2, 11, 14057$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T749

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -66, 253, -836, 2277, -5028, 9497, -15712, 21924, -24558, 21609, -15162, 8949, -4772, 2360, -1034, 383, -122, 35, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 35*x^18 - 122*x^17 + 383*x^16 - 1034*x^15 + 2360*x^14 - 4772*x^13 + 8949*x^12 - 15162*x^11 + 21609*x^10 - 24558*x^9 + 21924*x^8 - 15712*x^7 + 9497*x^6 - 5028*x^5 + 2277*x^4 - 836*x^3 + 253*x^2 - 66*x + 11)
 
gp: K = bnfinit(x^20 - 8*x^19 + 35*x^18 - 122*x^17 + 383*x^16 - 1034*x^15 + 2360*x^14 - 4772*x^13 + 8949*x^12 - 15162*x^11 + 21609*x^10 - 24558*x^9 + 21924*x^8 - 15712*x^7 + 9497*x^6 - 5028*x^5 + 2277*x^4 - 836*x^3 + 253*x^2 - 66*x + 11, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 35 x^{18} - 122 x^{17} + 383 x^{16} - 1034 x^{15} + 2360 x^{14} - 4772 x^{13} + 8949 x^{12} - 15162 x^{11} + 21609 x^{10} - 24558 x^{9} + 21924 x^{8} - 15712 x^{7} + 9497 x^{6} - 5028 x^{5} + 2277 x^{4} - 836 x^{3} + 253 x^{2} - 66 x + 11 \)

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:  \(677291336864611819508989952=2^{20}\cdot 11^{16}\cdot 14057\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.96$
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:  $2, 11, 14057$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{5913102250107823} a^{19} - \frac{2282974339465986}{5913102250107823} a^{18} + \frac{1624505686685077}{5913102250107823} a^{17} + \frac{2576859085857295}{5913102250107823} a^{16} - \frac{2618294341657397}{5913102250107823} a^{15} + \frac{305786764018017}{5913102250107823} a^{14} + \frac{555517779578785}{5913102250107823} a^{13} - \frac{1635294752383448}{5913102250107823} a^{12} + \frac{1527586954760343}{5913102250107823} a^{11} - \frac{2433369386134969}{5913102250107823} a^{10} + \frac{1841742726132853}{5913102250107823} a^{9} + \frac{685828065080453}{5913102250107823} a^{8} + \frac{1991671511553675}{5913102250107823} a^{7} - \frac{46091162681526}{257091402178601} a^{6} - \frac{998181749926445}{5913102250107823} a^{5} + \frac{1214289732393926}{5913102250107823} a^{4} + \frac{169209819491122}{5913102250107823} a^{3} + \frac{1127568888760057}{5913102250107823} a^{2} - \frac{2889148061002186}{5913102250107823} a + \frac{1057105054550597}{5913102250107823}$

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:  $9$
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:  \( 114550.655929 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T749:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n749 are not computed
Character table for t20n749 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.219503494144.2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
14057Data not computed