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Properties

Label 21.5.12435938466...1789.1
Degree $21$
Signature $[5, 8]$
Discriminant $1709^{9}$
Root discriminant $24.29$
Ramified prime $1709$
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![-1, -4, -23, 291, -1122, 2658, -4365, 5201, -4472, 2526, -415, -987, 1479, -1304, 819, -359, 90, 16, -36, 22, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 22*x^19 - 36*x^18 + 16*x^17 + 90*x^16 - 359*x^15 + 819*x^14 - 1304*x^13 + 1479*x^12 - 987*x^11 - 415*x^10 + 2526*x^9 - 4472*x^8 + 5201*x^7 - 4365*x^6 + 2658*x^5 - 1122*x^4 + 291*x^3 - 23*x^2 - 4*x - 1)
 
gp: K = bnfinit(x^21 - 7*x^20 + 22*x^19 - 36*x^18 + 16*x^17 + 90*x^16 - 359*x^15 + 819*x^14 - 1304*x^13 + 1479*x^12 - 987*x^11 - 415*x^10 + 2526*x^9 - 4472*x^8 + 5201*x^7 - 4365*x^6 + 2658*x^5 - 1122*x^4 + 291*x^3 - 23*x^2 - 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 22 x^{19} - 36 x^{18} + 16 x^{17} + 90 x^{16} - 359 x^{15} + 819 x^{14} - 1304 x^{13} + 1479 x^{12} - 987 x^{11} - 415 x^{10} + 2526 x^{9} - 4472 x^{8} + 5201 x^{7} - 4365 x^{6} + 2658 x^{5} - 1122 x^{4} + 291 x^{3} - 23 x^{2} - 4 x - 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(124359384669552020013669011789=1709^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.29$
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:  $1709$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{18} a^{19} - \frac{1}{9} a^{18} + \frac{1}{18} a^{17} - \frac{1}{2} a^{16} - \frac{2}{9} a^{15} + \frac{7}{18} a^{14} + \frac{4}{9} a^{13} + \frac{4}{9} a^{12} - \frac{1}{9} a^{11} - \frac{5}{18} a^{10} + \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{7}{18} a^{5} + \frac{7}{18} a^{3} - \frac{7}{18} a^{2} - \frac{1}{18} a - \frac{5}{18}$, $\frac{1}{36047662921891716066} a^{20} + \frac{196662763647717617}{12015887640630572022} a^{19} + \frac{189981485209077787}{4005295880210190674} a^{18} + \frac{5727521827357294}{18023831460945858033} a^{17} - \frac{629111572013849905}{36047662921891716066} a^{16} - \frac{11977334044957507699}{36047662921891716066} a^{15} + \frac{14739898299992585101}{36047662921891716066} a^{14} + \frac{1772941824975574853}{6007943820315286011} a^{13} - \frac{7790083093070146949}{18023831460945858033} a^{12} + \frac{4892674331918374867}{12015887640630572022} a^{11} - \frac{3242240226946434055}{36047662921891716066} a^{10} + \frac{908415634060942814}{6007943820315286011} a^{9} - \frac{207731062712792584}{6007943820315286011} a^{8} - \frac{7465698590950850506}{18023831460945858033} a^{7} + \frac{22418105644484437}{12015887640630572022} a^{6} + \frac{4467430223265061913}{36047662921891716066} a^{5} - \frac{2180434757552293211}{36047662921891716066} a^{4} + \frac{7890157768491887177}{18023831460945858033} a^{3} - \frac{2358700932798396515}{6007943820315286011} a^{2} + \frac{6801715316140220254}{18023831460945858033} a + \frac{2953810513831749557}{36047662921891716066}$

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:  $12$
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:  \( 5054661.69478 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
1709Data not computed