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Properties

Label 21.7.17715608156...9264.1
Degree $21$
Signature $[7, 7]$
Discriminant $-\,2^{36}\cdot 3^{18}\cdot 13^{16}$
Root discriminant $59.39$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3596, -36110, 119110, -179097, 104248, 57937, -152086, 128156, -46606, -17641, 29502, -14691, 7056, -4521, -48, 1741, -406, -226, 70, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 + 4*x^19 + 70*x^18 - 226*x^17 - 406*x^16 + 1741*x^15 - 48*x^14 - 4521*x^13 + 7056*x^12 - 14691*x^11 + 29502*x^10 - 17641*x^9 - 46606*x^8 + 128156*x^7 - 152086*x^6 + 57937*x^5 + 104248*x^4 - 179097*x^3 + 119110*x^2 - 36110*x + 3596)
 
gp: K = bnfinit(x^21 - 2*x^20 + 4*x^19 + 70*x^18 - 226*x^17 - 406*x^16 + 1741*x^15 - 48*x^14 - 4521*x^13 + 7056*x^12 - 14691*x^11 + 29502*x^10 - 17641*x^9 - 46606*x^8 + 128156*x^7 - 152086*x^6 + 57937*x^5 + 104248*x^4 - 179097*x^3 + 119110*x^2 - 36110*x + 3596, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} + 4 x^{19} + 70 x^{18} - 226 x^{17} - 406 x^{16} + 1741 x^{15} - 48 x^{14} - 4521 x^{13} + 7056 x^{12} - 14691 x^{11} + 29502 x^{10} - 17641 x^{9} - 46606 x^{8} + 128156 x^{7} - 152086 x^{6} + 57937 x^{5} + 104248 x^{4} - 179097 x^{3} + 119110 x^{2} - 36110 x + 3596 \)

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:  \(-17715608156920361474020168853580939264=-\,2^{36}\cdot 3^{18}\cdot 13^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.39$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} + \frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} + \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{5481050966965610957212527430421283377680523956} a^{20} + \frac{4066619980241105594799446414305323613183207}{2740525483482805478606263715210641688840261978} a^{19} + \frac{19099757517404530192353782573520788812961033}{210809652575600421431251055016203206833866306} a^{18} - \frac{668635474764590503727631201951976357351733127}{5481050966965610957212527430421283377680523956} a^{17} - \frac{1095131520156285401240054265371075448662142411}{5481050966965610957212527430421283377680523956} a^{16} + \frac{1076581318155571878017268972460534276594051957}{5481050966965610957212527430421283377680523956} a^{15} - \frac{1099834662941431003601670094435474228164050801}{5481050966965610957212527430421283377680523956} a^{14} - \frac{1326539463395470583869620122337026363968678917}{5481050966965610957212527430421283377680523956} a^{13} - \frac{269250037974066410782753826677708948149363663}{5481050966965610957212527430421283377680523956} a^{12} + \frac{2275023191019718402528973150842998417353736425}{5481050966965610957212527430421283377680523956} a^{11} + \frac{2048568015030561274310461386920829161635730671}{5481050966965610957212527430421283377680523956} a^{10} + \frac{1230655313563569274335150420559247097066854525}{5481050966965610957212527430421283377680523956} a^{9} + \frac{626806903988479230904786946549838579099268897}{5481050966965610957212527430421283377680523956} a^{8} - \frac{1625300006585198686084345193819041303386651553}{5481050966965610957212527430421283377680523956} a^{7} - \frac{78852573800449575830395726500006373386148669}{1370262741741402739303131857605320844420130989} a^{6} + \frac{1907148971435699098585043082293442310245244279}{5481050966965610957212527430421283377680523956} a^{5} + \frac{2205341870268609430226457889638289036503581333}{5481050966965610957212527430421283377680523956} a^{4} - \frac{230005239789924940031480598147747807922386904}{1370262741741402739303131857605320844420130989} a^{3} + \frac{482606628156818490330563749172946221588320079}{2740525483482805478606263715210641688840261978} a^{2} - \frac{474647293741206488782929029738673301189049496}{1370262741741402739303131857605320844420130989} a - \frac{476639608586646491957004902735859750313422878}{1370262741741402739303131857605320844420130989}$

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:  \( 260777898205 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.104.1, 7.7.138584369664.1

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 R $21$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.12.24.315$x^{12} + 32 x^{11} - 10 x^{10} + 8 x^{9} - 18 x^{8} + 32 x^{7} + 20 x^{6} + 24 x^{5} - 24 x^{4} + 32 x^{3} + 16 x^{2} - 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
3Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$