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Properties

Label 21.3.17608825670...3447.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 31^{14}$
Root discriminant $47.68$
Ramified primes $7, 31$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7785, 11237, 12672, 9951, 1703, -6715, -10148, -3301, 6188, 1560, 2660, 2308, -916, -420, -213, -353, -16, -3, -3, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 7*x^19 - 3*x^18 - 3*x^17 - 16*x^16 - 353*x^15 - 213*x^14 - 420*x^13 - 916*x^12 + 2308*x^11 + 2660*x^10 + 1560*x^9 + 6188*x^8 - 3301*x^7 - 10148*x^6 - 6715*x^5 + 1703*x^4 + 9951*x^3 + 12672*x^2 + 11237*x + 7785)
 
gp: K = bnfinit(x^21 + 7*x^19 - 3*x^18 - 3*x^17 - 16*x^16 - 353*x^15 - 213*x^14 - 420*x^13 - 916*x^12 + 2308*x^11 + 2660*x^10 + 1560*x^9 + 6188*x^8 - 3301*x^7 - 10148*x^6 - 6715*x^5 + 1703*x^4 + 9951*x^3 + 12672*x^2 + 11237*x + 7785, 1)
 

Normalized defining polynomial

\( x^{21} + 7 x^{19} - 3 x^{18} - 3 x^{17} - 16 x^{16} - 353 x^{15} - 213 x^{14} - 420 x^{13} - 916 x^{12} + 2308 x^{11} + 2660 x^{10} + 1560 x^{9} + 6188 x^{8} - 3301 x^{7} - 10148 x^{6} - 6715 x^{5} + 1703 x^{4} + 9951 x^{3} + 12672 x^{2} + 11237 x + 7785 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-176088256709912967722303227668133447=-\,7^{17}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.68$
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:  $7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{16} - \frac{1}{12} a^{14} + \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} + \frac{5}{12} a^{3} - \frac{1}{4} a^{2} - \frac{1}{6} a + \frac{1}{4}$, $\frac{1}{48} a^{18} - \frac{1}{48} a^{17} - \frac{1}{16} a^{16} - \frac{1}{12} a^{15} + \frac{5}{48} a^{14} + \frac{1}{12} a^{13} + \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{12} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{4} a^{6} + \frac{1}{6} a^{5} + \frac{23}{48} a^{4} - \frac{5}{16} a^{3} + \frac{7}{48} a^{2} + \frac{1}{4} a - \frac{7}{16}$, $\frac{1}{96} a^{19} - \frac{11}{96} a^{16} - \frac{11}{96} a^{15} + \frac{5}{96} a^{14} - \frac{1}{24} a^{13} + \frac{1}{6} a^{12} - \frac{1}{4} a^{11} - \frac{1}{24} a^{10} + \frac{1}{8} a^{9} - \frac{1}{6} a^{8} + \frac{1}{24} a^{7} - \frac{5}{24} a^{6} - \frac{17}{96} a^{5} - \frac{1}{12} a^{4} - \frac{3}{8} a^{3} + \frac{7}{96} a^{2} + \frac{19}{96} a + \frac{13}{32}$, $\frac{1}{31451904325177804057008707788474754346816} a^{20} - \frac{80931533289596842790796298916359969505}{31451904325177804057008707788474754346816} a^{19} + \frac{20340187496102993900687345790667629293}{7862976081294451014252176947118688586704} a^{18} - \frac{23697434575391634745811583113397269135}{1850112019128106121000512222851456138048} a^{17} - \frac{223582189501340231419838726016461526703}{7862976081294451014252176947118688586704} a^{16} + \frac{72855866276781606122012223715690405421}{982872010161806376781522118389836073338} a^{15} - \frac{3314515271224830677087524366538154016933}{31451904325177804057008707788474754346816} a^{14} - \frac{1058521728896154273141872484953402789863}{7862976081294451014252176947118688586704} a^{13} + \frac{634390163994059606657320675697287323373}{3931488040647225507126088473559344293352} a^{12} + \frac{460248757828029431520017314576326464905}{7862976081294451014252176947118688586704} a^{11} + \frac{37361352310069226217453499972476320545}{982872010161806376781522118389836073338} a^{10} + \frac{52917515318880279394112547631465367029}{7862976081294451014252176947118688586704} a^{9} + \frac{142725307645967122478957485967128555189}{7862976081294451014252176947118688586704} a^{8} + \frac{242496706273549247496936114988671983885}{1310496013549075169042029491186448097784} a^{7} - \frac{7121451362435152357173387864688390142317}{31451904325177804057008707788474754346816} a^{6} + \frac{5102439122876669419191619337775868604659}{10483968108392601352336235929491584782272} a^{5} + \frac{143919875842452905804058840934074070952}{491436005080903188390761059194918036669} a^{4} + \frac{10851599295185399923307144958837583829359}{31451904325177804057008707788474754346816} a^{3} + \frac{1612857288043299717481292427219204649319}{3931488040647225507126088473559344293352} a^{2} - \frac{46873755799283853238289517613041865367}{7862976081294451014252176947118688586704} a + \frac{1700676562846104765488266053322911513149}{3494656036130867117445411976497194927424}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 6975844064.607592 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.47089.2, 7.1.15521617447.1

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

Sibling fields

Degree 21 siblings: data not computed
Degree 42 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ R $21$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
7Data not computed
31Data not computed