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Properties

Label 21.3.20964804513...8623.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 37^{14}$
Root discriminant $53.65$
Ramified primes $7, 37$
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![343, -1029, -2401, 8232, -1029, -4802, 1372, 6321, -14945, 24304, -13769, 2989, -2044, 3122, -3220, 1785, -791, 322, -132, 41, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^20 + 41*x^19 - 132*x^18 + 322*x^17 - 791*x^16 + 1785*x^15 - 3220*x^14 + 3122*x^13 - 2044*x^12 + 2989*x^11 - 13769*x^10 + 24304*x^9 - 14945*x^8 + 6321*x^7 + 1372*x^6 - 4802*x^5 - 1029*x^4 + 8232*x^3 - 2401*x^2 - 1029*x + 343)
 
gp: K = bnfinit(x^21 - 9*x^20 + 41*x^19 - 132*x^18 + 322*x^17 - 791*x^16 + 1785*x^15 - 3220*x^14 + 3122*x^13 - 2044*x^12 + 2989*x^11 - 13769*x^10 + 24304*x^9 - 14945*x^8 + 6321*x^7 + 1372*x^6 - 4802*x^5 - 1029*x^4 + 8232*x^3 - 2401*x^2 - 1029*x + 343, 1)
 

Normalized defining polynomial

\( x^{21} - 9 x^{20} + 41 x^{19} - 132 x^{18} + 322 x^{17} - 791 x^{16} + 1785 x^{15} - 3220 x^{14} + 3122 x^{13} - 2044 x^{12} + 2989 x^{11} - 13769 x^{10} + 24304 x^{9} - 14945 x^{8} + 6321 x^{7} + 1372 x^{6} - 4802 x^{5} - 1029 x^{4} + 8232 x^{3} - 2401 x^{2} - 1029 x + 343 \)

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:  \(-2096480451371533622031253968968078623=-\,7^{17}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.65$
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:  $7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6}$, $\frac{1}{7} a^{10} + \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6}$, $\frac{1}{7} a^{11} + \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{6}$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{7}$, $\frac{1}{49} a^{15} - \frac{2}{49} a^{14} - \frac{1}{49} a^{13} + \frac{1}{49} a^{12} - \frac{1}{7} a^{8} - \frac{2}{7} a^{6}$, $\frac{1}{49} a^{16} + \frac{2}{49} a^{14} - \frac{1}{49} a^{13} + \frac{2}{49} a^{12} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6}$, $\frac{1}{49} a^{17} + \frac{3}{49} a^{14} - \frac{3}{49} a^{13} - \frac{2}{49} a^{12} + \frac{2}{7} a^{8} - \frac{3}{7} a^{6}$, $\frac{1}{49} a^{18} + \frac{3}{49} a^{14} + \frac{1}{49} a^{13} - \frac{3}{49} a^{12} - \frac{1}{7} a^{7} - \frac{3}{7} a^{6}$, $\frac{1}{49} a^{19} - \frac{3}{49} a^{12} + \frac{2}{7} a^{8} - \frac{1}{7} a^{6}$, $\frac{1}{792521755288394290756091304407628905} a^{20} + \frac{4512501483506918329750822925409748}{792521755288394290756091304407628905} a^{19} + \frac{2249947712854647435053125084499937}{792521755288394290756091304407628905} a^{18} - \frac{2959722912516313955346566646770583}{792521755288394290756091304407628905} a^{17} + \frac{1901972485440881266414422945661171}{792521755288394290756091304407628905} a^{16} - \frac{1024628257576180976158551459747842}{113217393612627755822298757772518415} a^{15} + \frac{21111518919152158996997052070186467}{792521755288394290756091304407628905} a^{14} - \frac{13177179621401352228084487736216851}{792521755288394290756091304407628905} a^{13} - \frac{3145460603557636028250124427889709}{158504351057678858151218260881525781} a^{12} - \frac{7383656007199472314798536387285677}{113217393612627755822298757772518415} a^{11} + \frac{3621369321977707299866157689090143}{113217393612627755822298757772518415} a^{10} + \frac{998314771897298895222606980102512}{16173913373232536546042679681788345} a^{9} - \frac{1268729540931985690859706464194273}{3234782674646507309208535936357669} a^{8} - \frac{6127567745954314665354732913102808}{22643478722525551164459751554503683} a^{7} - \frac{12397900185346802840434554897220092}{113217393612627755822298757772518415} a^{6} + \frac{2004201034838618411993913891206686}{16173913373232536546042679681788345} a^{5} - \frac{2508748572690600382983101260952391}{16173913373232536546042679681788345} a^{4} - \frac{6054358570344349076964332293708738}{16173913373232536546042679681788345} a^{3} - \frac{132221900060416289608343062916373}{16173913373232536546042679681788345} a^{2} - \frac{762227982061594052590860765609725}{3234782674646507309208535936357669} a + \frac{1430728779213639650150576355406394}{16173913373232536546042679681788345}$

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:  \( 3649149592.701335 \) (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

3.3.67081.2, 7.1.31499023927.1

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 $21$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$37$37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$