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Properties

Label 20.0.22762830184...3281.1
Degree $20$
Signature $[0, 10]$
Discriminant $8311^{8}$
Root discriminant $36.97$
Ramified prime $8311$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2\times A_5$ (as 20T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -1536, 2560, -3328, 3776, -4128, 3856, -3504, 2916, -2248, 1669, -1124, 729, -438, 241, -129, 59, -26, 10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 10*x^18 - 26*x^17 + 59*x^16 - 129*x^15 + 241*x^14 - 438*x^13 + 729*x^12 - 1124*x^11 + 1669*x^10 - 2248*x^9 + 2916*x^8 - 3504*x^7 + 3856*x^6 - 4128*x^5 + 3776*x^4 - 3328*x^3 + 2560*x^2 - 1536*x + 1024)
 
gp: K = bnfinit(x^20 - 3*x^19 + 10*x^18 - 26*x^17 + 59*x^16 - 129*x^15 + 241*x^14 - 438*x^13 + 729*x^12 - 1124*x^11 + 1669*x^10 - 2248*x^9 + 2916*x^8 - 3504*x^7 + 3856*x^6 - 4128*x^5 + 3776*x^4 - 3328*x^3 + 2560*x^2 - 1536*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 10 x^{18} - 26 x^{17} + 59 x^{16} - 129 x^{15} + 241 x^{14} - 438 x^{13} + 729 x^{12} - 1124 x^{11} + 1669 x^{10} - 2248 x^{9} + 2916 x^{8} - 3504 x^{7} + 3856 x^{6} - 4128 x^{5} + 3776 x^{4} - 3328 x^{3} + 2560 x^{2} - 1536 x + 1024 \)

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:  \(22762830184956456665916278633281=8311^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.97$
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:  $8311$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} - \frac{3}{8} a^{8} + \frac{3}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{8} a^{11} + \frac{7}{16} a^{10} - \frac{3}{16} a^{9} + \frac{3}{16} a^{8} - \frac{1}{2} a^{7} + \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{7}{16} a^{4} - \frac{3}{8} a^{3}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{16} a^{12} + \frac{7}{32} a^{11} - \frac{3}{32} a^{10} - \frac{13}{32} a^{9} - \frac{1}{4} a^{8} + \frac{1}{32} a^{7} + \frac{7}{16} a^{6} - \frac{7}{32} a^{5} + \frac{5}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{15} - \frac{1}{32} a^{13} + \frac{7}{64} a^{12} - \frac{3}{64} a^{11} + \frac{19}{64} a^{10} - \frac{1}{8} a^{9} - \frac{31}{64} a^{8} - \frac{9}{32} a^{7} - \frac{7}{64} a^{6} + \frac{5}{32} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{128} a^{17} - \frac{1}{128} a^{16} - \frac{1}{64} a^{14} + \frac{7}{128} a^{13} - \frac{3}{128} a^{12} + \frac{19}{128} a^{11} - \frac{1}{16} a^{10} + \frac{33}{128} a^{9} - \frac{9}{64} a^{8} - \frac{7}{128} a^{7} - \frac{27}{64} a^{6} + \frac{3}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{256} a^{18} - \frac{1}{256} a^{17} - \frac{1}{128} a^{15} + \frac{7}{256} a^{14} - \frac{3}{256} a^{13} + \frac{19}{256} a^{12} - \frac{1}{32} a^{11} + \frac{33}{256} a^{10} - \frac{9}{128} a^{9} - \frac{7}{256} a^{8} - \frac{27}{128} a^{7} - \frac{5}{16} a^{6} - \frac{1}{16} a^{5} + \frac{3}{16} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{49664} a^{19} - \frac{17}{49664} a^{18} - \frac{35}{12416} a^{17} - \frac{3}{24832} a^{16} - \frac{633}{49664} a^{15} + \frac{197}{49664} a^{14} - \frac{577}{49664} a^{13} - \frac{321}{12416} a^{12} + \frac{9005}{49664} a^{11} + \frac{1199}{24832} a^{10} - \frac{10563}{49664} a^{9} + \frac{6857}{24832} a^{8} - \frac{5851}{12416} a^{7} - \frac{609}{6208} a^{6} - \frac{637}{3104} a^{5} + \frac{31}{776} a^{4} + \frac{55}{388} a^{3} + \frac{87}{194} a^{2} - \frac{22}{97} a + \frac{14}{97}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 10106391.1313 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_5$ (as 20T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

10.10.4771040786343841.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
8311Data not computed