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Properties

Label 20.4.13047078908...2704.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{10}\cdot 3^{10}\cdot 11^{18}\cdot 197^{2}$
Root discriminant $35.96$
Ramified primes $2, 3, 11, 197$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T262

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1583, 13581, -44238, 82114, -107107, 123377, -123380, 95198, -56057, 22627, -4927, -1804, 3013, -1855, 986, -351, 176, -45, 15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 15*x^18 - 45*x^17 + 176*x^16 - 351*x^15 + 986*x^14 - 1855*x^13 + 3013*x^12 - 1804*x^11 - 4927*x^10 + 22627*x^9 - 56057*x^8 + 95198*x^7 - 123380*x^6 + 123377*x^5 - 107107*x^4 + 82114*x^3 - 44238*x^2 + 13581*x - 1583)
 
gp: K = bnfinit(x^20 - 4*x^19 + 15*x^18 - 45*x^17 + 176*x^16 - 351*x^15 + 986*x^14 - 1855*x^13 + 3013*x^12 - 1804*x^11 - 4927*x^10 + 22627*x^9 - 56057*x^8 + 95198*x^7 - 123380*x^6 + 123377*x^5 - 107107*x^4 + 82114*x^3 - 44238*x^2 + 13581*x - 1583, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 15 x^{18} - 45 x^{17} + 176 x^{16} - 351 x^{15} + 986 x^{14} - 1855 x^{13} + 3013 x^{12} - 1804 x^{11} - 4927 x^{10} + 22627 x^{9} - 56057 x^{8} + 95198 x^{7} - 123380 x^{6} + 123377 x^{5} - 107107 x^{4} + 82114 x^{3} - 44238 x^{2} + 13581 x - 1583 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13047078908784463218470268232704=2^{10}\cdot 3^{10}\cdot 11^{18}\cdot 197^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.96$
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:  $2, 3, 11, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{7685467766561120218959661374006723725729979634} a^{19} + \frac{489300835191040255772132457899756304188780372}{3842733883280560109479830687003361862864989817} a^{18} - \frac{1274783614400264187543580214664865006427450807}{7685467766561120218959661374006723725729979634} a^{17} + \frac{876212543996915472341939884721154863047206833}{7685467766561120218959661374006723725729979634} a^{16} - \frac{167694983172702908196706360601015216565018930}{3842733883280560109479830687003361862864989817} a^{15} + \frac{723964431634966943460688005775450156502453320}{3842733883280560109479830687003361862864989817} a^{14} + \frac{222115225632189505631657857857915704412451177}{7685467766561120218959661374006723725729979634} a^{13} + \frac{1463996188404909748286576608027928618690898518}{3842733883280560109479830687003361862864989817} a^{12} + \frac{1696308137771530455951798363361656119919180131}{7685467766561120218959661374006723725729979634} a^{11} - \frac{1168723671051241595788270507823782437188205782}{3842733883280560109479830687003361862864989817} a^{10} + \frac{738929573579687077145099832438728667369281193}{3842733883280560109479830687003361862864989817} a^{9} + \frac{484162317456508553154405467860790938605303211}{7685467766561120218959661374006723725729979634} a^{8} - \frac{1777410525918417871355410121142020698447358797}{3842733883280560109479830687003361862864989817} a^{7} + \frac{22047259902204169964288805502485496276801589}{7685467766561120218959661374006723725729979634} a^{6} - \frac{1411891070355329519013735230291827500027233490}{3842733883280560109479830687003361862864989817} a^{5} + \frac{541821341500608307795361029411665841230431977}{7685467766561120218959661374006723725729979634} a^{4} - \frac{3251127312902531667735799495622838112186234083}{7685467766561120218959661374006723725729979634} a^{3} + \frac{1711222046593386172120644982384310367862948074}{3842733883280560109479830687003361862864989817} a^{2} - \frac{2594538014438065705823232849923782093436466267}{7685467766561120218959661374006723725729979634} a - \frac{1704753901090074924401218071882578836814749714}{3842733883280560109479830687003361862864989817}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 17076745.3026 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T262:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2560
The 40 conjugacy class representatives for t20n262
Character table for t20n262 is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\)

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 32 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 R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{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$$[2, 2, 2, 2]^{5}$
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
197Data not computed