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Properties

Label 21.3.48725579790...0768.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,2^{18}\cdot 7^{17}\cdot 19^{14}$
Root discriminant $62.32$
Ramified primes $2, 7, 19$
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![3849502, -3898716, -2056094, 2159304, 518290, -382958, -91574, 99573, -114172, 15699, 48058, -26199, 2060, 4533, -2386, 231, 158, -103, 28, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 + 3*x^19 + 28*x^18 - 103*x^17 + 158*x^16 + 231*x^15 - 2386*x^14 + 4533*x^13 + 2060*x^12 - 26199*x^11 + 48058*x^10 + 15699*x^9 - 114172*x^8 + 99573*x^7 - 91574*x^6 - 382958*x^5 + 518290*x^4 + 2159304*x^3 - 2056094*x^2 - 3898716*x + 3849502)
 
gp: K = bnfinit(x^21 - 2*x^20 + 3*x^19 + 28*x^18 - 103*x^17 + 158*x^16 + 231*x^15 - 2386*x^14 + 4533*x^13 + 2060*x^12 - 26199*x^11 + 48058*x^10 + 15699*x^9 - 114172*x^8 + 99573*x^7 - 91574*x^6 - 382958*x^5 + 518290*x^4 + 2159304*x^3 - 2056094*x^2 - 3898716*x + 3849502, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} + 3 x^{19} + 28 x^{18} - 103 x^{17} + 158 x^{16} + 231 x^{15} - 2386 x^{14} + 4533 x^{13} + 2060 x^{12} - 26199 x^{11} + 48058 x^{10} + 15699 x^{9} - 114172 x^{8} + 99573 x^{7} - 91574 x^{6} - 382958 x^{5} + 518290 x^{4} + 2159304 x^{3} - 2056094 x^{2} - 3898716 x + 3849502 \)

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:  \(-48725579790519404577278592926363680768=-\,2^{18}\cdot 7^{17}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.32$
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:  $2, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{1}{7} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{14} a^{14} - \frac{1}{14} a^{13} - \frac{1}{14} a^{12} - \frac{5}{14} a^{11} - \frac{1}{2} a^{10} - \frac{5}{14} a^{9} - \frac{5}{14} a^{8} + \frac{1}{14} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{14} a^{15} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{1}{14} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2}$, $\frac{1}{14} a^{16} + \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{5}{14} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{14} a^{17} + \frac{1}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{14} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{98} a^{18} - \frac{3}{98} a^{17} + \frac{1}{98} a^{16} + \frac{1}{98} a^{15} - \frac{3}{49} a^{12} + \frac{1}{7} a^{11} - \frac{15}{98} a^{10} - \frac{3}{98} a^{9} + \frac{45}{98} a^{8} + \frac{5}{14} a^{7} - \frac{24}{49} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{10}{49} a^{3} - \frac{5}{49} a^{2} - \frac{24}{49} a + \frac{4}{49}$, $\frac{1}{490} a^{19} + \frac{1}{490} a^{18} + \frac{1}{49} a^{17} + \frac{6}{245} a^{16} - \frac{17}{490} a^{15} + \frac{4}{245} a^{13} + \frac{9}{245} a^{12} - \frac{211}{490} a^{11} + \frac{27}{70} a^{10} + \frac{97}{245} a^{9} - \frac{22}{245} a^{8} - \frac{237}{490} a^{7} + \frac{58}{245} a^{6} + \frac{12}{35} a^{5} - \frac{3}{245} a^{4} + \frac{53}{245} a^{3} - \frac{23}{245} a^{2} + \frac{18}{49} a + \frac{16}{245}$, $\frac{1}{30113832770364809235436502655044022846466834358494300} a^{20} + \frac{2583840965015941286594293785402333914960479262041}{4301976110052115605062357522149146120923833479784900} a^{19} + \frac{57485329695507419929856194790775632590015071312623}{15056916385182404617718251327522011423233417179247150} a^{18} + \frac{35480825336831412128422143570434961329018594678918}{7528458192591202308859125663761005711616708589623575} a^{17} + \frac{110675831048480513613342458953164999883782531406761}{6022766554072961847087300531008804569293366871698860} a^{16} + \frac{463002744794085980978788459577119353295148862703853}{30113832770364809235436502655044022846466834358494300} a^{15} + \frac{169561923405248029182555008078265133235429532160287}{7528458192591202308859125663761005711616708589623575} a^{14} + \frac{43543044530801846044617639152119295329091651900134}{7528458192591202308859125663761005711616708589623575} a^{13} - \frac{334225781576024619003190104230737524695346659342813}{30113832770364809235436502655044022846466834358494300} a^{12} + \frac{14189036960259301289916687408200629949359130959258153}{30113832770364809235436502655044022846466834358494300} a^{11} - \frac{1678045687712953675485812563311809358965761936645108}{7528458192591202308859125663761005711616708589623575} a^{10} - \frac{1093596474390745456360151327468805257843651833714099}{3011383277036480923543650265504402284646683435849430} a^{9} - \frac{359821973265629266055673747078311186112902935362123}{4301976110052115605062357522149146120923833479784900} a^{8} + \frac{5886604879419502362270131473891932663760202386009399}{30113832770364809235436502655044022846466834358494300} a^{7} - \frac{3723411139948377543254499490564683954857497053687983}{15056916385182404617718251327522011423233417179247150} a^{6} - \frac{3529131482800152708779979270878534526943390826181562}{7528458192591202308859125663761005711616708589623575} a^{5} + \frac{105222221702084973161987043205554712237031881785121}{430197611005211560506235752214914612092383347978490} a^{4} - \frac{551540717355317331810165256885884035155958955621184}{1505691638518240461771825132752201142323341717924715} a^{3} - \frac{2640186258576952770384613447656045465479938883316879}{7528458192591202308859125663761005711616708589623575} a^{2} + \frac{441336225581415286146183995302655880306000987067291}{15056916385182404617718251327522011423233417179247150} a - \frac{1609713462999897662298740838251415304790581421778459}{15056916385182404617718251327522011423233417179247150}$

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:  \( 179627315671.82587 \) (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

3.3.17689.2, 7.1.140179523008.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R $21$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
19Data not computed