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Properties

Label 21.21.2116651955...9097.1
Degree $21$
Signature $[21, 0]$
Discriminant $11^{7}\cdot 43^{19}$
Root discriminant $66.84$
Ramified primes $11, 43$
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![-1, 18, 82, -2084, -3790, 62865, 210641, -12040, -626935, -344830, 551344, 287628, -277632, -69748, 74059, 1551, -8463, 737, 395, -55, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - 55*x^19 + 395*x^18 + 737*x^17 - 8463*x^16 + 1551*x^15 + 74059*x^14 - 69748*x^13 - 277632*x^12 + 287628*x^11 + 551344*x^10 - 344830*x^9 - 626935*x^8 - 12040*x^7 + 210641*x^6 + 62865*x^5 - 3790*x^4 - 2084*x^3 + 82*x^2 + 18*x - 1)
 
gp: K = bnfinit(x^21 - 6*x^20 - 55*x^19 + 395*x^18 + 737*x^17 - 8463*x^16 + 1551*x^15 + 74059*x^14 - 69748*x^13 - 277632*x^12 + 287628*x^11 + 551344*x^10 - 344830*x^9 - 626935*x^8 - 12040*x^7 + 210641*x^6 + 62865*x^5 - 3790*x^4 - 2084*x^3 + 82*x^2 + 18*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} - 55 x^{19} + 395 x^{18} + 737 x^{17} - 8463 x^{16} + 1551 x^{15} + 74059 x^{14} - 69748 x^{13} - 277632 x^{12} + 287628 x^{11} + 551344 x^{10} - 344830 x^{9} - 626935 x^{8} - 12040 x^{7} + 210641 x^{6} + 62865 x^{5} - 3790 x^{4} - 2084 x^{3} + 82 x^{2} + 18 x - 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(211665195536810304505959087862660999097=11^{7}\cdot 43^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.84$
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:  $11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{19351} a^{19} + \frac{3373}{19351} a^{18} - \frac{7940}{19351} a^{17} + \frac{82}{19351} a^{16} + \frac{988}{19351} a^{15} + \frac{4783}{19351} a^{14} - \frac{6188}{19351} a^{13} + \frac{5962}{19351} a^{12} - \frac{1146}{19351} a^{11} - \frac{3793}{19351} a^{10} + \frac{9232}{19351} a^{9} + \frac{3418}{19351} a^{8} - \frac{5669}{19351} a^{7} - \frac{6421}{19351} a^{6} + \frac{2869}{19351} a^{5} - \frac{3790}{19351} a^{4} - \frac{8370}{19351} a^{3} - \frac{5260}{19351} a^{2} + \frac{1005}{19351} a - \frac{6161}{19351}$, $\frac{1}{58567930514408262071694525264253523} a^{20} + \frac{695789792243269397429868218240}{58567930514408262071694525264253523} a^{19} + \frac{22828799580426465235611394614753101}{58567930514408262071694525264253523} a^{18} + \frac{8620053342600641115351064327028125}{58567930514408262071694525264253523} a^{17} + \frac{26016331442348040716140590605408649}{58567930514408262071694525264253523} a^{16} + \frac{14926801475888888686112588452570423}{58567930514408262071694525264253523} a^{15} - \frac{23465345914954158789280954228283443}{58567930514408262071694525264253523} a^{14} + \frac{17625640687476414461475754289203767}{58567930514408262071694525264253523} a^{13} - \frac{3514971315707678644870019043440169}{58567930514408262071694525264253523} a^{12} + \frac{27389503394377254353011224286750644}{58567930514408262071694525264253523} a^{11} - \frac{1801951142641298719164421413501689}{58567930514408262071694525264253523} a^{10} - \frac{1655255000171251071927243472027884}{58567930514408262071694525264253523} a^{9} + \frac{21046115280375301252204924219931317}{58567930514408262071694525264253523} a^{8} + \frac{4893356605469690614033916661620786}{58567930514408262071694525264253523} a^{7} + \frac{9495115942320696945171460474612811}{58567930514408262071694525264253523} a^{6} + \frac{8279297738897250659977763877586462}{58567930514408262071694525264253523} a^{5} - \frac{2926602977867947696542067269745421}{58567930514408262071694525264253523} a^{4} + \frac{26999434596468338162022258888318672}{58567930514408262071694525264253523} a^{3} - \frac{1693571651994303900770034606318262}{58567930514408262071694525264253523} a^{2} + \frac{24541331214103142991627213843269719}{58567930514408262071694525264253523} a - \frac{10890392316148990939124449323150363}{58567930514408262071694525264253523}$

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:  $20$
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:  \( 2240296145980 \) (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

3.3.473.1, 7.7.6321363049.1

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 $21$ ${\href{/LocalNumberField/3.14.0.1}{14} }{,}\,{\href{/LocalNumberField/3.7.0.1}{7} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }$ R $21$ $21$ $21$

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
$11$11.7.0.1$x^{7} - x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
11.14.7.2$x^{14} - 1771561 x^{2} + 77948684$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.14.13.1$x^{14} - 43$$14$$1$$13$$C_{14}$$[\ ]_{14}$