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Properties

Label 21.7.46797540717...2416.1
Degree $21$
Signature $[7, 7]$
Discriminant $-\,2^{20}\cdot 11^{7}\cdot 73^{12}$
Root discriminant $49.96$
Ramified primes $2, 11, 73$
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, 360, 204, -4776, 9552, -14508, 16832, -11568, 6420, -68, -2916, 2404, -2088, 630, -262, 24, 60, -6, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 16*x^19 - 6*x^18 + 60*x^17 + 24*x^16 - 262*x^15 + 630*x^14 - 2088*x^13 + 2404*x^12 - 2916*x^11 - 68*x^10 + 6420*x^9 - 11568*x^8 + 16832*x^7 - 14508*x^6 + 9552*x^5 - 4776*x^4 + 204*x^3 + 360*x^2 - 600*x + 500)
 
gp: K = bnfinit(x^21 + 16*x^19 - 6*x^18 + 60*x^17 + 24*x^16 - 262*x^15 + 630*x^14 - 2088*x^13 + 2404*x^12 - 2916*x^11 - 68*x^10 + 6420*x^9 - 11568*x^8 + 16832*x^7 - 14508*x^6 + 9552*x^5 - 4776*x^4 + 204*x^3 + 360*x^2 - 600*x + 500, 1)
 

Normalized defining polynomial

\( x^{21} + 16 x^{19} - 6 x^{18} + 60 x^{17} + 24 x^{16} - 262 x^{15} + 630 x^{14} - 2088 x^{13} + 2404 x^{12} - 2916 x^{11} - 68 x^{10} + 6420 x^{9} - 11568 x^{8} + 16832 x^{7} - 14508 x^{6} + 9552 x^{5} - 4776 x^{4} + 204 x^{3} + 360 x^{2} - 600 x + 500 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-467975407174598390024135662204092416=-\,2^{20}\cdot 11^{7}\cdot 73^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.96$
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:  $2, 11, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{2} a^{16}$, $\frac{1}{2} a^{17}$, $\frac{1}{2} a^{18}$, $\frac{1}{10} a^{19} + \frac{1}{10} a^{17} - \frac{1}{10} a^{16} - \frac{1}{10} a^{14} - \frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{328780762255263217654479996404203450} a^{20} - \frac{448227912444602187964759941968381}{65756152451052643530895999280840690} a^{19} + \frac{27812704464966233669807447273205333}{164390381127631608827239998202101725} a^{18} + \frac{11008537446580600697008143949892557}{164390381127631608827239998202101725} a^{17} + \frac{7618556180002207908680019155636563}{65756152451052643530895999280840690} a^{16} + \frac{8809539598746008888014305103389449}{328780762255263217654479996404203450} a^{15} - \frac{24068129715172893079757946510234291}{164390381127631608827239998202101725} a^{14} + \frac{4593647111914314845563277846669544}{32878076225526321765447999640420345} a^{13} - \frac{88161158878320599348278940106863}{328780762255263217654479996404203450} a^{12} + \frac{29722592719342295315448064465444222}{164390381127631608827239998202101725} a^{11} + \frac{36907388251118027891732668027679532}{164390381127631608827239998202101725} a^{10} - \frac{5712964316354640736159125175046644}{164390381127631608827239998202101725} a^{9} - \frac{8794831376104464917433123855133084}{32878076225526321765447999640420345} a^{8} - \frac{64527802554553761384596107125007334}{164390381127631608827239998202101725} a^{7} - \frac{34945015073208023132935981850172989}{164390381127631608827239998202101725} a^{6} + \frac{58924302247291319321567171861992916}{164390381127631608827239998202101725} a^{5} + \frac{69207120654078858972978385220784896}{164390381127631608827239998202101725} a^{4} + \frac{79272953646576141716454193934695382}{164390381127631608827239998202101725} a^{3} + \frac{36476317099526836778685955741181592}{164390381127631608827239998202101725} a^{2} + \frac{887672643125245362258855004727119}{32878076225526321765447999640420345} a + \frac{312599306797944974277879905623288}{6575615245105264353089599928084069}$

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:  \( 11785083997.5 \) (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

3.1.44.1, 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 $21$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
73Data not computed