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Properties

Label 20.0.47172009111...1376.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 3^{4}\cdot 23^{2}\cdot 401^{10}$
Root discriminant $48.27$
Ramified primes $2, 3, 23, 401$
Class number $384$ (GRH)
Class group $[2, 2, 96]$ (GRH)
Galois group 20T347

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![240003, -45198, 258894, -149958, 219405, -69594, 185033, -39686, 55041, -25204, 21804, -10070, 5559, -2402, 839, -374, 193, -106, 37, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 37*x^18 - 106*x^17 + 193*x^16 - 374*x^15 + 839*x^14 - 2402*x^13 + 5559*x^12 - 10070*x^11 + 21804*x^10 - 25204*x^9 + 55041*x^8 - 39686*x^7 + 185033*x^6 - 69594*x^5 + 219405*x^4 - 149958*x^3 + 258894*x^2 - 45198*x + 240003)
 
gp: K = bnfinit(x^20 - 8*x^19 + 37*x^18 - 106*x^17 + 193*x^16 - 374*x^15 + 839*x^14 - 2402*x^13 + 5559*x^12 - 10070*x^11 + 21804*x^10 - 25204*x^9 + 55041*x^8 - 39686*x^7 + 185033*x^6 - 69594*x^5 + 219405*x^4 - 149958*x^3 + 258894*x^2 - 45198*x + 240003, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 37 x^{18} - 106 x^{17} + 193 x^{16} - 374 x^{15} + 839 x^{14} - 2402 x^{13} + 5559 x^{12} - 10070 x^{11} + 21804 x^{10} - 25204 x^{9} + 55041 x^{8} - 39686 x^{7} + 185033 x^{6} - 69594 x^{5} + 219405 x^{4} - 149958 x^{3} + 258894 x^{2} - 45198 x + 240003 \)

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:  \(4717200911168296461448196340581376=2^{10}\cdot 3^{4}\cdot 23^{2}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.27$
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:  $2, 3, 23, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{15} - \frac{1}{3} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{6} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{17} + \frac{1}{18} a^{16} - \frac{1}{9} a^{15} - \frac{1}{18} a^{14} + \frac{1}{18} a^{13} - \frac{1}{9} a^{12} + \frac{4}{9} a^{11} + \frac{7}{18} a^{10} - \frac{1}{2} a^{9} + \frac{7}{18} a^{8} + \frac{1}{3} a^{7} - \frac{7}{18} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} - \frac{7}{18} a^{3} - \frac{1}{6} a$, $\frac{1}{18} a^{18} + \frac{2}{9} a^{15} - \frac{2}{9} a^{14} - \frac{1}{3} a^{13} - \frac{5}{18} a^{12} - \frac{7}{18} a^{11} + \frac{4}{9} a^{10} + \frac{1}{18} a^{9} + \frac{4}{9} a^{8} + \frac{4}{9} a^{7} - \frac{5}{18} a^{6} + \frac{7}{18} a^{5} - \frac{5}{18} a^{4} + \frac{1}{18} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{1964941498161264254938051152044599882625979694240067502} a^{19} + \frac{4738045490057692047655280979927765002849239244393161}{1964941498161264254938051152044599882625979694240067502} a^{18} + \frac{45782774068857766486700521691992634036381469525085615}{1964941498161264254938051152044599882625979694240067502} a^{17} - \frac{62052826707415455398300849406088811555947151846666677}{1964941498161264254938051152044599882625979694240067502} a^{16} - \frac{227847504653008401276045296303280875266493076281083798}{982470749080632127469025576022299941312989847120033751} a^{15} + \frac{679630890373341783059735560386422276699384846023217341}{1964941498161264254938051152044599882625979694240067502} a^{14} + \frac{107801406075809793872084813044235598008877495576298000}{982470749080632127469025576022299941312989847120033751} a^{13} + \frac{308111567404257478110899608475360725912052761971198872}{982470749080632127469025576022299941312989847120033751} a^{12} - \frac{276718235190402463346232693585555974874891800800042075}{654980499387088084979350384014866627541993231413355834} a^{11} - \frac{410660531615621861466599633554032003190043730027260974}{982470749080632127469025576022299941312989847120033751} a^{10} - \frac{92855455298494926434658622689287018587788761458263400}{327490249693544042489675192007433313770996615706677917} a^{9} + \frac{108290722119972899031674232419821127879841836482260371}{1964941498161264254938051152044599882625979694240067502} a^{8} - \frac{45452904800339994683291042418109911277823503980011971}{218326833129029361659783461338288875847331077137785278} a^{7} + \frac{164330143060683197366114129062352140490856187975291723}{1964941498161264254938051152044599882625979694240067502} a^{6} + \frac{355378715521422967804387030947579883209459089636541253}{982470749080632127469025576022299941312989847120033751} a^{5} - \frac{24747587249744246877620109298542595061304406875614512}{327490249693544042489675192007433313770996615706677917} a^{4} + \frac{96395479424015708648608757268644541501027340924999699}{327490249693544042489675192007433313770996615706677917} a^{3} - \frac{43754610200366615138552345150533939676385948369184375}{218326833129029361659783461338288875847331077137785278} a^{2} - \frac{34161572798088456594972350530437484624943207896167831}{72775611043009787219927820446096291949110359045928426} a + \frac{19894218105955443069428184976955592516864199857156655}{72775611043009787219927820446096291949110359045928426}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{96}$, which has order $384$ (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:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T347:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 104 conjugacy class representatives for t20n347 are not computed
Character table for t20n347 is not computed

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5, 10.10.10368641602001.1

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 R R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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}$
2.10.10.3$x^{10} - 9 x^{8} + 22 x^{6} - 46 x^{4} + 9 x^{2} - 9$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.2$x^{4} - 23 x^{2} + 3703$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
401Data not computed