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Properties

Label 20.4.38698352640...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{38}\cdot 3^{10}\cdot 5^{22}$
Root discriminant $37.97$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![176419, 369370, -1027855, 1691310, -1837185, 1346964, -714330, 233820, -65565, 4090, -1103, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 - 1103*x^10 + 4090*x^9 - 65565*x^8 + 233820*x^7 - 714330*x^6 + 1346964*x^5 - 1837185*x^4 + 1691310*x^3 - 1027855*x^2 + 369370*x + 176419)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 - 1103*x^10 + 4090*x^9 - 65565*x^8 + 233820*x^7 - 714330*x^6 + 1346964*x^5 - 1837185*x^4 + 1691310*x^3 - 1027855*x^2 + 369370*x + 176419, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 30 x^{17} - 105 x^{16} + 228 x^{15} + 90 x^{14} - 540 x^{13} + 45 x^{12} + 770 x^{11} - 1103 x^{10} + 4090 x^{9} - 65565 x^{8} + 233820 x^{7} - 714330 x^{6} + 1346964 x^{5} - 1837185 x^{4} + 1691310 x^{3} - 1027855 x^{2} + 369370 x + 176419 \)

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:  \(38698352640000000000000000000000=2^{38}\cdot 3^{10}\cdot 5^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.97$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{9} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{45} a^{10} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{2}{15} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} + \frac{8}{45}$, $\frac{1}{45} a^{11} - \frac{1}{45} a^{6} - \frac{1}{9} a^{5} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{4}{15} a - \frac{4}{9}$, $\frac{1}{135} a^{12} - \frac{1}{27} a^{9} - \frac{2}{45} a^{7} - \frac{2}{27} a^{6} + \frac{1}{9} a^{4} - \frac{2}{27} a^{3} + \frac{7}{15} a^{2} + \frac{1}{9} a - \frac{10}{27}$, $\frac{1}{135} a^{13} + \frac{1}{135} a^{10} - \frac{2}{45} a^{8} + \frac{4}{27} a^{7} + \frac{1}{9} a^{6} - \frac{7}{45} a^{5} + \frac{4}{27} a^{4} + \frac{1}{45} a^{3} + \frac{4}{9} a^{2} - \frac{10}{27} a + \frac{1}{45}$, $\frac{1}{135} a^{14} + \frac{1}{135} a^{11} - \frac{2}{45} a^{9} + \frac{1}{27} a^{8} - \frac{1}{9} a^{7} - \frac{2}{45} a^{6} - \frac{2}{27} a^{5} - \frac{4}{45} a^{4} + \frac{1}{3} a^{3} - \frac{7}{27} a^{2} - \frac{4}{45} a - \frac{4}{9}$, $\frac{1}{135} a^{15} - \frac{1}{27} a^{9} - \frac{1}{9} a^{6} - \frac{2}{15} a^{5} + \frac{13}{27} a^{3} + \frac{1}{3} a + \frac{8}{135}$, $\frac{1}{2025} a^{16} + \frac{7}{2025} a^{15} + \frac{2}{2025} a^{14} + \frac{2}{675} a^{13} - \frac{4}{2025} a^{12} + \frac{2}{2025} a^{11} + \frac{2}{405} a^{10} - \frac{4}{675} a^{9} + \frac{64}{2025} a^{8} - \frac{67}{675} a^{7} + \frac{2}{405} a^{6} - \frac{56}{2025} a^{5} - \frac{184}{2025} a^{4} - \frac{28}{225} a^{3} + \frac{218}{2025} a^{2} + \frac{14}{2025} a - \frac{209}{2025}$, $\frac{1}{2025} a^{17} - \frac{2}{2025} a^{15} + \frac{7}{2025} a^{14} - \frac{1}{2025} a^{13} + \frac{11}{2025} a^{11} + \frac{8}{2025} a^{10} - \frac{17}{2025} a^{9} + \frac{56}{2025} a^{8} + \frac{247}{2025} a^{7} + \frac{103}{675} a^{6} - \frac{167}{2025} a^{5} - \frac{269}{2025} a^{4} - \frac{283}{2025} a^{3} - \frac{34}{675} a^{2} + \frac{413}{2025} a + \frac{443}{2025}$, $\frac{1}{194435150074454475} a^{18} - \frac{1}{21603905563828275} a^{17} + \frac{15958467430384}{64811716691484825} a^{16} - \frac{14185304382556}{7201301854609425} a^{15} - \frac{143384593225522}{64811716691484825} a^{14} - \frac{122729939320079}{64811716691484825} a^{13} - \frac{160469079467}{65137403710035} a^{12} + \frac{585301793381579}{64811716691484825} a^{11} - \frac{157901724956083}{64811716691484825} a^{10} + \frac{7775975455995371}{194435150074454475} a^{9} - \frac{3488514535902527}{64811716691484825} a^{8} - \frac{1687044407913526}{64811716691484825} a^{7} - \frac{3685397242174751}{64811716691484825} a^{6} + \frac{328863502198682}{12962343338296965} a^{5} - \frac{1760113314810326}{64811716691484825} a^{4} + \frac{30971082222303959}{64811716691484825} a^{3} - \frac{14538719596147114}{64811716691484825} a^{2} - \frac{705894851875943}{2400433951536475} a + \frac{73028095908324032}{194435150074454475}$, $\frac{1}{96155764682670641612025} a^{19} + \frac{16484}{6410384312178042774135} a^{18} - \frac{22055907410181558241}{96155764682670641612025} a^{17} + \frac{3125342273310949678}{32051921560890213870675} a^{16} - \frac{191498911348162040809}{96155764682670641612025} a^{15} - \frac{304596411670373752564}{96155764682670641612025} a^{14} - \frac{107453213850486316733}{96155764682670641612025} a^{13} - \frac{7894972685508658904}{6410384312178042774135} a^{12} - \frac{628581776906583087497}{96155764682670641612025} a^{11} + \frac{169897474465867162528}{32051921560890213870675} a^{10} - \frac{317214659835891995344}{96155764682670641612025} a^{9} - \frac{4925182674836710466732}{96155764682670641612025} a^{8} + \frac{10053397884276376530908}{96155764682670641612025} a^{7} + \frac{21843541713268894426}{237421641191779362005} a^{6} - \frac{10600219797934893006427}{96155764682670641612025} a^{5} + \frac{2007977864474949295436}{96155764682670641612025} a^{4} - \frac{41547869933297498888384}{96155764682670641612025} a^{3} - \frac{1300165519974438769847}{10683973853630071290225} a^{2} - \frac{2305462106442058832873}{32051921560890213870675} a + \frac{5892507569566103329613}{19231152936534128322405}$

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:  $11$
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:  \( 142510796.64703235 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{6})\), 5.1.50000.1, 10.2.1244160000000000.14, 10.2.6220800000000000.10, 10.2.12500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$