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Properties

Label 21.21.2279745164...6768.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{20}\cdot 37^{7}\cdot 73^{12}$
Root discriminant $74.85$
Ramified primes $2, 37, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times C_7:C_3$ (as 21T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![500, 600, -7560, -11796, 32136, 52528, -67836, -106488, 83328, 117252, -61308, -75324, 26324, 28968, -6230, -6550, 720, 812, -30, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 48*x^19 - 30*x^18 + 812*x^17 + 720*x^16 - 6550*x^15 - 6230*x^14 + 28968*x^13 + 26324*x^12 - 75324*x^11 - 61308*x^10 + 117252*x^9 + 83328*x^8 - 106488*x^7 - 67836*x^6 + 52528*x^5 + 32136*x^4 - 11796*x^3 - 7560*x^2 + 600*x + 500)
 
gp: K = bnfinit(x^21 - 48*x^19 - 30*x^18 + 812*x^17 + 720*x^16 - 6550*x^15 - 6230*x^14 + 28968*x^13 + 26324*x^12 - 75324*x^11 - 61308*x^10 + 117252*x^9 + 83328*x^8 - 106488*x^7 - 67836*x^6 + 52528*x^5 + 32136*x^4 - 11796*x^3 - 7560*x^2 + 600*x + 500, 1)
 

Normalized defining polynomial

\( x^{21} - 48 x^{19} - 30 x^{18} + 812 x^{17} + 720 x^{16} - 6550 x^{15} - 6230 x^{14} + 28968 x^{13} + 26324 x^{12} - 75324 x^{11} - 61308 x^{10} + 117252 x^{9} + 83328 x^{8} - 106488 x^{7} - 67836 x^{6} + 52528 x^{5} + 32136 x^{4} - 11796 x^{3} - 7560 x^{2} + 600 x + 500 \)

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:  \(2279745164404038997779147090610663456768=2^{20}\cdot 37^{7}\cdot 73^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.85$
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:  $2, 37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{10} a^{19} + \frac{1}{5} a^{17} + \frac{1}{5} a^{15} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{2184966537498234641619950} a^{20} - \frac{6138363421119890898692}{218496653749823464161995} a^{19} + \frac{265481381505271662195727}{2184966537498234641619950} a^{18} + \frac{89946251832247263443811}{436993307499646928323990} a^{17} + \frac{156608199680499387256106}{1092483268749117320809975} a^{16} + \frac{69484912469423890416901}{436993307499646928323990} a^{15} - \frac{3918700823051121767738}{43699330749964692832399} a^{14} - \frac{24719129586541602940258}{218496653749823464161995} a^{13} - \frac{56320322147675810469107}{2184966537498234641619950} a^{12} - \frac{244257851317393191569568}{1092483268749117320809975} a^{11} + \frac{206534139887923957073998}{1092483268749117320809975} a^{10} - \frac{536742966930716292161789}{1092483268749117320809975} a^{9} + \frac{133502936104622391061681}{1092483268749117320809975} a^{8} + \frac{99303569434828937501169}{1092483268749117320809975} a^{7} - \frac{503221027991283596399799}{1092483268749117320809975} a^{6} - \frac{57226748401723101770613}{1092483268749117320809975} a^{5} + \frac{187167180033350004286524}{1092483268749117320809975} a^{4} - \frac{2970763342308067031262}{1092483268749117320809975} a^{3} - \frac{260270979950110783326983}{1092483268749117320809975} a^{2} + \frac{66732783003705733741151}{218496653749823464161995} a - \frac{7477153427526123941255}{43699330749964692832399}$

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:  \( 20438598341400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times C_7:C_3$ (as 21T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 15 conjugacy class representatives for $S_3\times C_7:C_3$
Character table for $S_3\times C_7:C_3$

Intermediate fields

3.3.148.1, 7.7.1817487424.1

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

Sibling fields

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 $21$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
37Data not computed
$73$73.3.0.1$x^{3} - x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
73.9.6.1$x^{9} + 3066 x^{6} + 3128123 x^{3} + 1067462648$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
73.9.6.1$x^{9} + 3066 x^{6} + 3128123 x^{3} + 1067462648$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$