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Properties

Label 21.21.4873621493...0000.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{49}\cdot 3^{8}\cdot 5^{8}\cdot 491^{8}$
Root discriminant $149.83$
Ramified primes $2, 3, 5, 491$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $SO(3,7)$ (as 21T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![53833824, -111827424, -275865888, 688442080, -93152336, -627347440, 313874448, 212267312, -172066922, -24499166, 43117146, -2098218, -5819847, 854237, 440572, -92372, -18268, 4720, 378, -114, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 3*x^20 - 114*x^19 + 378*x^18 + 4720*x^17 - 18268*x^16 - 92372*x^15 + 440572*x^14 + 854237*x^13 - 5819847*x^12 - 2098218*x^11 + 43117146*x^10 - 24499166*x^9 - 172066922*x^8 + 212267312*x^7 + 313874448*x^6 - 627347440*x^5 - 93152336*x^4 + 688442080*x^3 - 275865888*x^2 - 111827424*x + 53833824)
 
gp: K = bnfinit(x^21 - 3*x^20 - 114*x^19 + 378*x^18 + 4720*x^17 - 18268*x^16 - 92372*x^15 + 440572*x^14 + 854237*x^13 - 5819847*x^12 - 2098218*x^11 + 43117146*x^10 - 24499166*x^9 - 172066922*x^8 + 212267312*x^7 + 313874448*x^6 - 627347440*x^5 - 93152336*x^4 + 688442080*x^3 - 275865888*x^2 - 111827424*x + 53833824, 1)
 

Normalized defining polynomial

\( x^{21} - 3 x^{20} - 114 x^{19} + 378 x^{18} + 4720 x^{17} - 18268 x^{16} - 92372 x^{15} + 440572 x^{14} + 854237 x^{13} - 5819847 x^{12} - 2098218 x^{11} + 43117146 x^{10} - 24499166 x^{9} - 172066922 x^{8} + 212267312 x^{7} + 313874448 x^{6} - 627347440 x^{5} - 93152336 x^{4} + 688442080 x^{3} - 275865888 x^{2} - 111827424 x + 53833824 \)

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:  \(4873621493405139175690041549663712051200000000=2^{49}\cdot 3^{8}\cdot 5^{8}\cdot 491^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $149.83$
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:  $2, 3, 5, 491$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{12} - \frac{1}{24} a^{9} + \frac{5}{24} a^{8} - \frac{1}{12} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{5}{12} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{1}{12} a^{9} - \frac{1}{8} a^{8} - \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{6} a^{5} + \frac{5}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{48} a^{15} - \frac{1}{48} a^{13} - \frac{1}{48} a^{11} + \frac{1}{12} a^{10} + \frac{1}{16} a^{9} - \frac{1}{6} a^{8} - \frac{1}{12} a^{6} + \frac{11}{24} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a$, $\frac{1}{144} a^{16} + \frac{1}{144} a^{15} - \frac{1}{48} a^{14} + \frac{1}{144} a^{13} - \frac{5}{144} a^{12} + \frac{1}{48} a^{11} + \frac{1}{16} a^{10} - \frac{5}{48} a^{9} - \frac{1}{6} a^{8} + \frac{1}{9} a^{7} - \frac{13}{72} a^{6} - \frac{1}{24} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{144} a^{17} - \frac{1}{144} a^{15} - \frac{1}{72} a^{14} - \frac{1}{48} a^{13} - \frac{1}{36} a^{12} + \frac{1}{48} a^{11} - \frac{1}{24} a^{10} + \frac{1}{24} a^{9} + \frac{1}{36} a^{8} - \frac{1}{8} a^{7} - \frac{1}{9} a^{6} + \frac{11}{36} a^{5} - \frac{1}{3} a^{4} + \frac{1}{9} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{864} a^{18} + \frac{1}{864} a^{17} + \frac{1}{864} a^{16} + \frac{5}{864} a^{15} + \frac{1}{864} a^{14} + \frac{1}{864} a^{13} - \frac{11}{864} a^{12} - \frac{1}{32} a^{11} - \frac{1}{144} a^{10} - \frac{19}{432} a^{9} - \frac{91}{432} a^{8} - \frac{37}{432} a^{7} + \frac{1}{72} a^{6} - \frac{73}{216} a^{5} + \frac{49}{108} a^{4} + \frac{49}{108} a^{3} - \frac{4}{9} a^{2} - \frac{5}{18} a + \frac{1}{3}$, $\frac{1}{864} a^{19} - \frac{1}{432} a^{16} + \frac{1}{108} a^{15} - \frac{1}{48} a^{14} + \frac{7}{432} a^{12} - \frac{5}{288} a^{11} + \frac{11}{432} a^{10} + \frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{5}{432} a^{7} + \frac{53}{216} a^{6} - \frac{7}{24} a^{5} - \frac{2}{9} a^{4} + \frac{5}{108} a^{3} + \frac{1}{6} a^{2} + \frac{5}{18} a - \frac{1}{3}$, $\frac{1}{5692096291029920795706139477548174165994368} a^{20} - \frac{4159811315683046415288181079563983511}{355756018189370049731633717346760885374648} a^{19} - \frac{525432870055071438807680074272447326077}{2846048145514960397853069738774087082997184} a^{18} + \frac{2642313451409258525305580275622025544493}{1423024072757480198926534869387043541498592} a^{17} - \frac{2341819856392768162712099424197433930091}{1423024072757480198926534869387043541498592} a^{16} - \frac{60259593415742022230502756284343229519}{9882111616371390270323158815187802371518} a^{15} + \frac{15469605504427325345534013010668554742859}{1423024072757480198926534869387043541498592} a^{14} + \frac{1414270930379804982366577730163353680601}{118585339396456683243877905782253628458216} a^{13} + \frac{56098049452573468108565201469954214537661}{5692096291029920795706139477548174165994368} a^{12} + \frac{12047515587657006089623868631400857268501}{355756018189370049731633717346760885374648} a^{11} + \frac{325757061217064206817822152866770799733375}{2846048145514960397853069738774087082997184} a^{10} + \frac{82170953807456969857430854402442476680475}{1423024072757480198926534869387043541498592} a^{9} - \frac{206822718229189281391873562460189042050923}{948682715171653465951023246258029027665728} a^{8} + \frac{2424059690295900508758706226749956395603}{44469502273671256216454214668345110671831} a^{7} - \frac{38675315154358585923010809732883576348553}{355756018189370049731633717346760885374648} a^{6} + \frac{120060953189490849943483988162034901735691}{355756018189370049731633717346760885374648} a^{5} - \frac{32060866453407544163177403450176651697863}{118585339396456683243877905782253628458216} a^{4} + \frac{35187459211173661265170173221538512431915}{88939004547342512432908429336690221343662} a^{3} - \frac{4559688748050964067111982294649864981231}{19764223232742780540646317630375604743036} a^{2} + \frac{3096703104588961924381494011383351541731}{14823167424557085405484738222781703557277} a - \frac{607770203206711939349449084349463716341}{19764223232742780540646317630375604743036}$

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:  \( 2430053091190000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SO(3,7)$ (as 21T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 336
The 9 conjugacy class representatives for $SO(3,7)$
Character table for $SO(3,7)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 8 sibling: data not computed
Degree 14 sibling: data not computed
Degree 16 sibling: data not computed
Degree 24 sibling: data not computed
Degree 28 siblings: data not computed
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 R ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.10.6$x^{4} + 6 x^{2} + 3$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
2.4.9.7$x^{4} + 2 x^{2} + 6$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.8.22.83$x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
491Data not computed