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Properties

Label 20.0.35796303426...7609.1
Degree $20$
Signature $[0, 10]$
Discriminant $11^{9}\cdot 19^{15}$
Root discriminant $26.77$
Ramified primes $11, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1103, 1486, 1737, 721, 1411, 1180, 136, -616, -875, 362, 697, -47, -104, 164, 56, -6, -1, -2, 9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 9*x^18 - 2*x^17 - x^16 - 6*x^15 + 56*x^14 + 164*x^13 - 104*x^12 - 47*x^11 + 697*x^10 + 362*x^9 - 875*x^8 - 616*x^7 + 136*x^6 + 1180*x^5 + 1411*x^4 + 721*x^3 + 1737*x^2 + 1486*x + 1103)
 
gp: K = bnfinit(x^20 - x^19 + 9*x^18 - 2*x^17 - x^16 - 6*x^15 + 56*x^14 + 164*x^13 - 104*x^12 - 47*x^11 + 697*x^10 + 362*x^9 - 875*x^8 - 616*x^7 + 136*x^6 + 1180*x^5 + 1411*x^4 + 721*x^3 + 1737*x^2 + 1486*x + 1103, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 9 x^{18} - 2 x^{17} - x^{16} - 6 x^{15} + 56 x^{14} + 164 x^{13} - 104 x^{12} - 47 x^{11} + 697 x^{10} + 362 x^{9} - 875 x^{8} - 616 x^{7} + 136 x^{6} + 1180 x^{5} + 1411 x^{4} + 721 x^{3} + 1737 x^{2} + 1486 x + 1103 \)

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:  \(35796303426870968668217777609=11^{9}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.77$
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:  $11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{4545297625843460418864065705334963256537} a^{19} - \frac{1237406550083830108675531941816543702896}{4545297625843460418864065705334963256537} a^{18} + \frac{2063754047038686235518751854485696621920}{4545297625843460418864065705334963256537} a^{17} - \frac{1156987757922671741733671105757148670823}{4545297625843460418864065705334963256537} a^{16} - \frac{2011732729093566936609554736515429008727}{4545297625843460418864065705334963256537} a^{15} - \frac{1727043986650376568260240682944250188430}{4545297625843460418864065705334963256537} a^{14} + \frac{910181032579110211110147401290003507721}{4545297625843460418864065705334963256537} a^{13} + \frac{1947347504823520828150080671096070181108}{4545297625843460418864065705334963256537} a^{12} - \frac{1962546598329582572963075455917828755530}{4545297625843460418864065705334963256537} a^{11} - \frac{1841205155404534607461637916176975609548}{4545297625843460418864065705334963256537} a^{10} + \frac{2022723917846254517768252478301422398025}{4545297625843460418864065705334963256537} a^{9} - \frac{1057333461883384975212453083790297913028}{4545297625843460418864065705334963256537} a^{8} + \frac{1144796352156649274942031449842331313709}{4545297625843460418864065705334963256537} a^{7} + \frac{2139054587493032306842903111944734072163}{4545297625843460418864065705334963256537} a^{6} + \frac{429571217061833693291975984520913511659}{4545297625843460418864065705334963256537} a^{5} - \frac{1103739786966620870525441330313971699543}{4545297625843460418864065705334963256537} a^{4} + \frac{111679817243869293228231368537717172589}{4545297625843460418864065705334963256537} a^{3} + \frac{1939386515547870343984094924999223807952}{4545297625843460418864065705334963256537} a^{2} - \frac{1894125309582044280689775736327583248252}{4545297625843460418864065705334963256537} a - \frac{27374826053911520476337268792002866970}{4545297625843460418864065705334963256537}$

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:  \( 635235.6127553139 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.75449.1, 10.0.36252565459.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ $20$ $20$

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
$11$11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11.10.9.4$x^{10} - 99$$10$$1$$9$$C_{10}$$[\ ]_{10}$
19Data not computed