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Properties

Label 20.4.31427919143...8329.7
Degree $20$
Signature $[4, 8]$
Discriminant $3^{6}\cdot 401^{11}$
Root discriminant $37.57$
Ramified primes $3, 401$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T350

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 4536, 15561, 34776, 56742, 72231, 74687, 64085, 42080, 20593, 3776, -3405, -3418, -1378, -80, 249, 94, 4, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 8*x^18 + 4*x^17 + 94*x^16 + 249*x^15 - 80*x^14 - 1378*x^13 - 3418*x^12 - 3405*x^11 + 3776*x^10 + 20593*x^9 + 42080*x^8 + 64085*x^7 + 74687*x^6 + 72231*x^5 + 56742*x^4 + 34776*x^3 + 15561*x^2 + 4536*x + 729)
 
gp: K = bnfinit(x^20 - 4*x^19 - 8*x^18 + 4*x^17 + 94*x^16 + 249*x^15 - 80*x^14 - 1378*x^13 - 3418*x^12 - 3405*x^11 + 3776*x^10 + 20593*x^9 + 42080*x^8 + 64085*x^7 + 74687*x^6 + 72231*x^5 + 56742*x^4 + 34776*x^3 + 15561*x^2 + 4536*x + 729, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 8 x^{18} + 4 x^{17} + 94 x^{16} + 249 x^{15} - 80 x^{14} - 1378 x^{13} - 3418 x^{12} - 3405 x^{11} + 3776 x^{10} + 20593 x^{9} + 42080 x^{8} + 64085 x^{7} + 74687 x^{6} + 72231 x^{5} + 56742 x^{4} + 34776 x^{3} + 15561 x^{2} + 4536 x + 729 \)

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:  \(31427919143590467585816658408329=3^{6}\cdot 401^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.57$
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:  $3, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{3} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{18} - \frac{1}{18} a^{17} + \frac{1}{18} a^{16} + \frac{1}{18} a^{15} + \frac{1}{18} a^{14} - \frac{1}{6} a^{13} + \frac{1}{18} a^{12} - \frac{2}{9} a^{11} - \frac{1}{18} a^{10} - \frac{1}{3} a^{9} + \frac{1}{9} a^{8} - \frac{5}{18} a^{7} + \frac{5}{18} a^{6} + \frac{5}{18} a^{5} - \frac{7}{18} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{554597143926404983304534532435906} a^{19} - \frac{2944709314186405092761229351239}{277298571963202491652267266217953} a^{18} + \frac{29928811169149821464275520996995}{554597143926404983304534532435906} a^{17} - \frac{20830841504414552947252624094527}{277298571963202491652267266217953} a^{16} - \frac{7167923958356081785648762538983}{277298571963202491652267266217953} a^{15} - \frac{16901856929635937214055370666965}{184865714642134994434844844145302} a^{14} + \frac{86837373101345075054905229821225}{554597143926404983304534532435906} a^{13} - \frac{9544436757779025410408276524211}{277298571963202491652267266217953} a^{12} + \frac{162391163507763835072577948171399}{554597143926404983304534532435906} a^{11} + \frac{37759245753222403650352633377719}{184865714642134994434844844145302} a^{10} + \frac{118825411591921628010629539710581}{554597143926404983304534532435906} a^{9} + \frac{93717387177194474016143929768166}{277298571963202491652267266217953} a^{8} - \frac{20669139116092728775287121027709}{277298571963202491652267266217953} a^{7} + \frac{126402290055509127800669152463428}{277298571963202491652267266217953} a^{6} + \frac{118807671256935880071721257758635}{277298571963202491652267266217953} a^{5} + \frac{16079230584180933886173367515643}{92432857321067497217422422072651} a^{4} + \frac{6580952479359633009401425313699}{184865714642134994434844844145302} a^{3} + \frac{4173184923644514033257234303885}{20540634960237221603871649349478} a^{2} - \frac{16725259131745244828040830587859}{61621904880711664811614948048434} a - \frac{1253136141596219810535485996854}{3423439160039536933978608224913}$

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:  $11$
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:  \( 23826998.0564 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T350:

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 t20n350 are not computed
Character table for t20n350 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$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}$
401Data not computed