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Properties

Label 20.0.15914835482...9433.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 17^{15}$
Root discriminant $72.46$
Ramified primes $11, 17$
Class number $8194$ (GRH)
Class group $[8194]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83519437, 20544384, 26015925, 9962701, 8994354, 2390937, 2845143, -32881, 700706, -168519, 130294, -40054, 20558, -4874, 2986, -472, 348, -31, 27, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 27*x^18 - 31*x^17 + 348*x^16 - 472*x^15 + 2986*x^14 - 4874*x^13 + 20558*x^12 - 40054*x^11 + 130294*x^10 - 168519*x^9 + 700706*x^8 - 32881*x^7 + 2845143*x^6 + 2390937*x^5 + 8994354*x^4 + 9962701*x^3 + 26015925*x^2 + 20544384*x + 83519437)
 
gp: K = bnfinit(x^20 - x^19 + 27*x^18 - 31*x^17 + 348*x^16 - 472*x^15 + 2986*x^14 - 4874*x^13 + 20558*x^12 - 40054*x^11 + 130294*x^10 - 168519*x^9 + 700706*x^8 - 32881*x^7 + 2845143*x^6 + 2390937*x^5 + 8994354*x^4 + 9962701*x^3 + 26015925*x^2 + 20544384*x + 83519437, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 27 x^{18} - 31 x^{17} + 348 x^{16} - 472 x^{15} + 2986 x^{14} - 4874 x^{13} + 20558 x^{12} - 40054 x^{11} + 130294 x^{10} - 168519 x^{9} + 700706 x^{8} - 32881 x^{7} + 2845143 x^{6} + 2390937 x^{5} + 8994354 x^{4} + 9962701 x^{3} + 26015925 x^{2} + 20544384 x + 83519437 \)

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:  \(15914835482628690354834740122825579433=11^{18}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.46$
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:  $11, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(187=11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{187}(1,·)$, $\chi_{187}(67,·)$, $\chi_{187}(69,·)$, $\chi_{187}(135,·)$, $\chi_{187}(72,·)$, $\chi_{187}(137,·)$, $\chi_{187}(140,·)$, $\chi_{187}(13,·)$, $\chi_{187}(16,·)$, $\chi_{187}(21,·)$, $\chi_{187}(86,·)$, $\chi_{187}(152,·)$, $\chi_{187}(30,·)$, $\chi_{187}(98,·)$, $\chi_{187}(103,·)$, $\chi_{187}(169,·)$, $\chi_{187}(106,·)$, $\chi_{187}(183,·)$, $\chi_{187}(123,·)$, $\chi_{187}(149,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{30887} a^{11} + \frac{11}{30887} a^{9} + \frac{44}{30887} a^{7} + \frac{77}{30887} a^{5} + \frac{55}{30887} a^{3} + \frac{11}{30887} a + \frac{14665}{30887}$, $\frac{1}{30887} a^{12} + \frac{11}{30887} a^{10} + \frac{44}{30887} a^{8} + \frac{77}{30887} a^{6} + \frac{55}{30887} a^{4} + \frac{11}{30887} a^{2} + \frac{14665}{30887} a$, $\frac{1}{30887} a^{13} - \frac{77}{30887} a^{9} - \frac{407}{30887} a^{7} - \frac{792}{30887} a^{5} - \frac{594}{30887} a^{3} + \frac{14665}{30887} a^{2} - \frac{121}{30887} a - \frac{6880}{30887}$, $\frac{1}{30887} a^{14} - \frac{77}{30887} a^{10} - \frac{407}{30887} a^{8} - \frac{792}{30887} a^{6} - \frac{594}{30887} a^{4} + \frac{14665}{30887} a^{3} - \frac{121}{30887} a^{2} - \frac{6880}{30887} a$, $\frac{1}{61774} a^{15} - \frac{1}{61774} a^{13} - \frac{1}{61774} a^{12} + \frac{15438}{30887} a^{10} - \frac{15185}{30887} a^{9} - \frac{22}{30887} a^{8} + \frac{3003}{61774} a^{7} - \frac{77}{61774} a^{6} + \frac{6127}{61774} a^{5} - \frac{16277}{61774} a^{4} - \frac{26179}{61774} a^{3} + \frac{9331}{61774} a^{2} - \frac{13697}{61774} a + \frac{24153}{61774}$, $\frac{1}{2502643205086} a^{16} - \frac{8372121}{2502643205086} a^{15} - \frac{38655455}{2502643205086} a^{14} + \frac{18497203}{1251321602543} a^{13} + \frac{28251677}{2502643205086} a^{12} + \frac{17102227}{1251321602543} a^{11} - \frac{161013994131}{1251321602543} a^{10} - \frac{541055715937}{1251321602543} a^{9} + \frac{1218489227347}{2502643205086} a^{8} - \frac{466725833993}{1251321602543} a^{7} - \frac{534465219149}{1251321602543} a^{6} - \frac{257990473602}{1251321602543} a^{5} - \frac{380266742709}{1251321602543} a^{4} + \frac{541726175575}{1251321602543} a^{3} + \frac{67473101116}{1251321602543} a^{2} - \frac{53033734868}{1251321602543} a + \frac{771452120321}{2502643205086}$, $\frac{1}{2502643205086} a^{17} - \frac{5587193}{2502643205086} a^{15} + \frac{29313715}{2502643205086} a^{14} + \frac{13754234}{1251321602543} a^{13} - \frac{14442998}{1251321602543} a^{12} - \frac{10163137}{1251321602543} a^{11} + \frac{388769291034}{1251321602543} a^{10} - \frac{1131462396497}{2502643205086} a^{9} + \frac{301946858209}{2502643205086} a^{8} - \frac{1152499577741}{2502643205086} a^{7} + \frac{282522920791}{2502643205086} a^{6} + \frac{1044374861419}{2502643205086} a^{5} - \frac{895913484533}{2502643205086} a^{4} - \frac{184068791665}{2502643205086} a^{3} - \frac{277154437689}{2502643205086} a^{2} + \frac{114140262026}{1251321602543} a + \frac{538442389146}{1251321602543}$, $\frac{1}{2502643205086} a^{18} + \frac{748541}{2502643205086} a^{15} - \frac{33433677}{2502643205086} a^{14} + \frac{1784145}{2502643205086} a^{13} - \frac{11187486}{1251321602543} a^{12} - \frac{6331065}{1251321602543} a^{11} + \frac{852715945333}{2502643205086} a^{10} - \frac{653896134655}{2502643205086} a^{9} + \frac{454585492758}{1251321602543} a^{8} - \frac{107368331880}{1251321602543} a^{7} - \frac{82706047459}{1251321602543} a^{6} + \frac{471139620551}{1251321602543} a^{5} - \frac{8179119867}{1251321602543} a^{4} - \frac{127564951569}{1251321602543} a^{3} - \frac{1206291573573}{2502643205086} a^{2} - \frac{716328211553}{2502643205086} a - \frac{342673681438}{1251321602543}$, $\frac{1}{2502643205086} a^{19} - \frac{7690924}{1251321602543} a^{15} - \frac{941029}{1251321602543} a^{14} + \frac{2003665}{1251321602543} a^{13} + \frac{14791057}{2502643205086} a^{12} - \frac{18927955}{2502643205086} a^{11} - \frac{1057310949435}{2502643205086} a^{10} + \frac{63415729885}{1251321602543} a^{9} + \frac{155000634833}{2502643205086} a^{8} - \frac{586995306289}{1251321602543} a^{7} + \frac{368816411947}{1251321602543} a^{6} - \frac{459350571569}{1251321602543} a^{5} - \frac{593839740802}{1251321602543} a^{4} - \frac{1005857634667}{2502643205086} a^{3} + \frac{647531067881}{2502643205086} a^{2} - \frac{456704347033}{1251321602543} a + \frac{179121068921}{2502643205086}$

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

Class group and class number

$C_{8194}$, which has order $8194$ (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:  \( 3338983.62101 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.594473.1, \(\Q(\zeta_{11})^+\), 10.10.304358957700017.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ $20$ $20$ $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
11Data not computed
17Data not computed