↑

Properties

Label 20.2.35028156925...9072.2
Degree $20$
Signature $[2, 9]$
Discriminant $-\,2^{20}\cdot 11^{16}\cdot 727$
Root discriminant $18.93$
Ramified primes $2, 11, 727$
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, 22, -44, 44, -110, 318, -688, 1014, -1111, 1074, -1276, 1774, -2033, 1604, -754, 92, 140, -114, 45, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 45*x^18 - 114*x^17 + 140*x^16 + 92*x^15 - 754*x^14 + 1604*x^13 - 2033*x^12 + 1774*x^11 - 1276*x^10 + 1074*x^9 - 1111*x^8 + 1014*x^7 - 688*x^6 + 318*x^5 - 110*x^4 + 44*x^3 - 44*x^2 + 22*x - 11)
 
gp: K = bnfinit(x^20 - 10*x^19 + 45*x^18 - 114*x^17 + 140*x^16 + 92*x^15 - 754*x^14 + 1604*x^13 - 2033*x^12 + 1774*x^11 - 1276*x^10 + 1074*x^9 - 1111*x^8 + 1014*x^7 - 688*x^6 + 318*x^5 - 110*x^4 + 44*x^3 - 44*x^2 + 22*x - 11, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 45 x^{18} - 114 x^{17} + 140 x^{16} + 92 x^{15} - 754 x^{14} + 1604 x^{13} - 2033 x^{12} + 1774 x^{11} - 1276 x^{10} + 1074 x^{9} - 1111 x^{8} + 1014 x^{7} - 688 x^{6} + 318 x^{5} - 110 x^{4} + 44 x^{3} - 44 x^{2} + 22 x - 11 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-35028156925416005746819072=-\,2^{20}\cdot 11^{16}\cdot 727\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.93$
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:  $2, 11, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4689430281286673207} a^{19} + \frac{508747848322990240}{4689430281286673207} a^{18} + \frac{40800366806034554}{4689430281286673207} a^{17} - \frac{535491314997371782}{4689430281286673207} a^{16} - \frac{2001775049248533140}{4689430281286673207} a^{15} + \frac{1042832873037213740}{4689430281286673207} a^{14} - \frac{702171780034135912}{4689430281286673207} a^{13} - \frac{127903419945558659}{4689430281286673207} a^{12} - \frac{690947426590127543}{4689430281286673207} a^{11} + \frac{1684690046228558483}{4689430281286673207} a^{10} + \frac{1092244160319740111}{4689430281286673207} a^{9} + \frac{1528385538874524727}{4689430281286673207} a^{8} - \frac{522862929721501163}{4689430281286673207} a^{7} - \frac{1925991812044994359}{4689430281286673207} a^{6} - \frac{984779373732830823}{4689430281286673207} a^{5} - \frac{123533167178558608}{4689430281286673207} a^{4} + \frac{2159186882929712179}{4689430281286673207} a^{3} + \frac{1929360967972264820}{4689430281286673207} a^{2} + \frac{24580933642855137}{4689430281286673207} a + \frac{565655458414926656}{4689430281286673207}$

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:  $10$
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:  \( 43814.9305076 \) (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

\(\Q(\zeta_{11})^+\), 10.6.219503494144.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
2Data not computed
$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}$
727Data not computed