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Properties

Label 21.7.13347832346...6103.1
Degree $21$
Signature $[7, 7]$
Discriminant $-\,3^{7}\cdot 29^{19}$
Root discriminant $30.35$
Ramified primes $3, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7\times S_3$ (as 21T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-41, 711, -3057, -2184, 19798, -16215, -21283, 34796, -5328, -9164, -1214, 2315, 1991, -1692, 384, 324, -211, 28, 10, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 9*x^19 + 10*x^18 + 28*x^17 - 211*x^16 + 324*x^15 + 384*x^14 - 1692*x^13 + 1991*x^12 + 2315*x^11 - 1214*x^10 - 9164*x^9 - 5328*x^8 + 34796*x^7 - 21283*x^6 - 16215*x^5 + 19798*x^4 - 2184*x^3 - 3057*x^2 + 711*x - 41)
 
gp: K = bnfinit(x^21 - x^20 - 9*x^19 + 10*x^18 + 28*x^17 - 211*x^16 + 324*x^15 + 384*x^14 - 1692*x^13 + 1991*x^12 + 2315*x^11 - 1214*x^10 - 9164*x^9 - 5328*x^8 + 34796*x^7 - 21283*x^6 - 16215*x^5 + 19798*x^4 - 2184*x^3 - 3057*x^2 + 711*x - 41, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 9 x^{19} + 10 x^{18} + 28 x^{17} - 211 x^{16} + 324 x^{15} + 384 x^{14} - 1692 x^{13} + 1991 x^{12} + 2315 x^{11} - 1214 x^{10} - 9164 x^{9} - 5328 x^{8} + 34796 x^{7} - 21283 x^{6} - 16215 x^{5} + 19798 x^{4} - 2184 x^{3} - 3057 x^{2} + 711 x - 41 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-13347832346292311387708944226103=-\,3^{7}\cdot 29^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.35$
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:  $3, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{41} a^{15} - \frac{14}{41} a^{14} + \frac{20}{41} a^{13} - \frac{2}{41} a^{12} - \frac{2}{41} a^{11} - \frac{18}{41} a^{10} + \frac{13}{41} a^{9} - \frac{11}{41} a^{8} + \frac{15}{41} a^{7} - \frac{13}{41} a^{6} + \frac{15}{41} a^{5} + \frac{12}{41} a^{4} + \frac{2}{41} a^{3} - \frac{18}{41} a$, $\frac{1}{41} a^{16} - \frac{12}{41} a^{14} - \frac{9}{41} a^{13} + \frac{11}{41} a^{12} - \frac{5}{41} a^{11} + \frac{7}{41} a^{10} + \frac{7}{41} a^{9} - \frac{16}{41} a^{8} - \frac{8}{41} a^{7} - \frac{3}{41} a^{6} + \frac{17}{41} a^{5} + \frac{6}{41} a^{4} - \frac{13}{41} a^{3} - \frac{18}{41} a^{2} - \frac{6}{41} a$, $\frac{1}{41} a^{17} - \frac{13}{41} a^{14} + \frac{5}{41} a^{13} + \frac{12}{41} a^{12} - \frac{17}{41} a^{11} - \frac{4}{41} a^{10} + \frac{17}{41} a^{9} - \frac{17}{41} a^{8} + \frac{13}{41} a^{7} - \frac{16}{41} a^{6} - \frac{19}{41} a^{5} + \frac{8}{41} a^{4} + \frac{6}{41} a^{3} - \frac{6}{41} a^{2} - \frac{11}{41} a$, $\frac{1}{41} a^{18} - \frac{13}{41} a^{14} - \frac{15}{41} a^{13} - \frac{2}{41} a^{12} + \frac{11}{41} a^{11} - \frac{12}{41} a^{10} - \frac{12}{41} a^{9} - \frac{7}{41} a^{8} + \frac{15}{41} a^{7} + \frac{17}{41} a^{6} - \frac{2}{41} a^{5} - \frac{2}{41} a^{4} + \frac{20}{41} a^{3} - \frac{11}{41} a^{2} + \frac{12}{41} a$, $\frac{1}{41} a^{19} + \frac{8}{41} a^{14} + \frac{12}{41} a^{13} - \frac{15}{41} a^{12} + \frac{3}{41} a^{11} - \frac{2}{41} a^{9} - \frac{5}{41} a^{8} + \frac{7}{41} a^{7} - \frac{7}{41} a^{6} - \frac{12}{41} a^{5} + \frac{12}{41} a^{4} + \frac{15}{41} a^{3} + \frac{12}{41} a^{2} + \frac{12}{41} a$, $\frac{1}{365563799184390541497995162065545955799} a^{20} - \frac{1776999270702097912359924674636854732}{365563799184390541497995162065545955799} a^{19} + \frac{4228711311334994834608408365495602852}{365563799184390541497995162065545955799} a^{18} + \frac{2861593344425352373586600460792592126}{365563799184390541497995162065545955799} a^{17} - \frac{1274798571521345190959222018525723704}{365563799184390541497995162065545955799} a^{16} + \frac{2200670510997224679006515228810069431}{365563799184390541497995162065545955799} a^{15} + \frac{52824015022055143680422552124722292760}{365563799184390541497995162065545955799} a^{14} - \frac{131650121961001651014395231598812813828}{365563799184390541497995162065545955799} a^{13} - \frac{164204834393031154861557971100885355214}{365563799184390541497995162065545955799} a^{12} + \frac{25423088297481639129073503427556826755}{365563799184390541497995162065545955799} a^{11} + \frac{49858553592851114973264068653850579280}{365563799184390541497995162065545955799} a^{10} - \frac{25023426337938327464486692890191357538}{365563799184390541497995162065545955799} a^{9} + \frac{157396572160357961624248312168099164281}{365563799184390541497995162065545955799} a^{8} + \frac{138396505426880684252780716405008413837}{365563799184390541497995162065545955799} a^{7} + \frac{78887663111406784708608528067517710581}{365563799184390541497995162065545955799} a^{6} - \frac{64734706597477419830190802439452721837}{365563799184390541497995162065545955799} a^{5} + \frac{103758077087866981108295034759345776655}{365563799184390541497995162065545955799} a^{4} + \frac{18554932622691161613226065733281559781}{365563799184390541497995162065545955799} a^{3} + \frac{131072072777462277791260352512244653023}{365563799184390541497995162065545955799} a^{2} + \frac{25146333915967143800952471146343495082}{365563799184390541497995162065545955799} a - \frac{2115148945441356319646095591183052524}{8916190224009525402390125904037706239}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 35353252.3989 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7\times S_3$ (as 21T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 21 conjugacy class representatives for $C_7\times S_3$
Character table for $C_7\times S_3$ is not computed

Intermediate fields

3.1.87.1, 7.7.594823321.1

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

Sibling fields

Galois closure: 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 $21$ R ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ $21$ $21$ $21$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }$ R ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ $21$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{7}$

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$3.7.0.1$x^{7} + x^{2} - x + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_29.2t1.1c1$1$ $ 3 \cdot 29 $ $x^{2} - x + 22$ $C_2$ (as 2T1) $1$ $-1$
1.3_29.14t1.1c1$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.1c1$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
1.3_29.14t1.1c2$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
1.3_29.14t1.1c3$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.1c2$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.29.7t1.1c3$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
1.3_29.14t1.1c4$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.1c4$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.29.7t1.1c5$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
1.3_29.14t1.1c5$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
1.3_29.14t1.1c6$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.1c6$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 2.3_29.3t2.1c1$2$ $ 3 \cdot 29 $ $x^{3} - x^{2} + 2 x + 1$ $S_3$ (as 3T2) $1$ $0$
* 2.3_29e2.21t6.1c1$2$ $ 3 \cdot 29^{2}$ $x^{21} - x^{20} - 9 x^{19} + 10 x^{18} + 28 x^{17} - 211 x^{16} + 324 x^{15} + 384 x^{14} - 1692 x^{13} + 1991 x^{12} + 2315 x^{11} - 1214 x^{10} - 9164 x^{9} - 5328 x^{8} + 34796 x^{7} - 21283 x^{6} - 16215 x^{5} + 19798 x^{4} - 2184 x^{3} - 3057 x^{2} + 711 x - 41$ $C_7\times S_3$ (as 21T6) $0$ $0$
* 2.3_29e2.21t6.1c2$2$ $ 3 \cdot 29^{2}$ $x^{21} - x^{20} - 9 x^{19} + 10 x^{18} + 28 x^{17} - 211 x^{16} + 324 x^{15} + 384 x^{14} - 1692 x^{13} + 1991 x^{12} + 2315 x^{11} - 1214 x^{10} - 9164 x^{9} - 5328 x^{8} + 34796 x^{7} - 21283 x^{6} - 16215 x^{5} + 19798 x^{4} - 2184 x^{3} - 3057 x^{2} + 711 x - 41$ $C_7\times S_3$ (as 21T6) $0$ $0$
* 2.3_29e2.21t6.1c3$2$ $ 3 \cdot 29^{2}$ $x^{21} - x^{20} - 9 x^{19} + 10 x^{18} + 28 x^{17} - 211 x^{16} + 324 x^{15} + 384 x^{14} - 1692 x^{13} + 1991 x^{12} + 2315 x^{11} - 1214 x^{10} - 9164 x^{9} - 5328 x^{8} + 34796 x^{7} - 21283 x^{6} - 16215 x^{5} + 19798 x^{4} - 2184 x^{3} - 3057 x^{2} + 711 x - 41$ $C_7\times S_3$ (as 21T6) $0$ $0$
* 2.3_29e2.21t6.1c4$2$ $ 3 \cdot 29^{2}$ $x^{21} - x^{20} - 9 x^{19} + 10 x^{18} + 28 x^{17} - 211 x^{16} + 324 x^{15} + 384 x^{14} - 1692 x^{13} + 1991 x^{12} + 2315 x^{11} - 1214 x^{10} - 9164 x^{9} - 5328 x^{8} + 34796 x^{7} - 21283 x^{6} - 16215 x^{5} + 19798 x^{4} - 2184 x^{3} - 3057 x^{2} + 711 x - 41$ $C_7\times S_3$ (as 21T6) $0$ $0$
* 2.3_29e2.21t6.1c5$2$ $ 3 \cdot 29^{2}$ $x^{21} - x^{20} - 9 x^{19} + 10 x^{18} + 28 x^{17} - 211 x^{16} + 324 x^{15} + 384 x^{14} - 1692 x^{13} + 1991 x^{12} + 2315 x^{11} - 1214 x^{10} - 9164 x^{9} - 5328 x^{8} + 34796 x^{7} - 21283 x^{6} - 16215 x^{5} + 19798 x^{4} - 2184 x^{3} - 3057 x^{2} + 711 x - 41$ $C_7\times S_3$ (as 21T6) $0$ $0$
* 2.3_29e2.21t6.1c6$2$ $ 3 \cdot 29^{2}$ $x^{21} - x^{20} - 9 x^{19} + 10 x^{18} + 28 x^{17} - 211 x^{16} + 324 x^{15} + 384 x^{14} - 1692 x^{13} + 1991 x^{12} + 2315 x^{11} - 1214 x^{10} - 9164 x^{9} - 5328 x^{8} + 34796 x^{7} - 21283 x^{6} - 16215 x^{5} + 19798 x^{4} - 2184 x^{3} - 3057 x^{2} + 711 x - 41$ $C_7\times S_3$ (as 21T6) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.