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Properties

Label 21.11.1521363284...3867.1
Degree $21$
Signature $[11, 5]$
Discriminant $-\,7^{15}\cdot 223\cdot 80317\cdot 1789163334752592959$
Root discriminant $65.80$
Ramified primes $7, 223, 80317, 1789163334752592959$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T159

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -9, 9, 200, 312, -744, -2312, -438, 4200, 2992, -3705, -3915, 1730, 2493, -438, -895, 57, 186, -3, -21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 57*x^16 - 895*x^15 - 438*x^14 + 2493*x^13 + 1730*x^12 - 3915*x^11 - 3705*x^10 + 2992*x^9 + 4200*x^8 - 438*x^7 - 2312*x^6 - 744*x^5 + 312*x^4 + 200*x^3 + 9*x^2 - 9*x - 1)
 
gp: K = bnfinit(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 57*x^16 - 895*x^15 - 438*x^14 + 2493*x^13 + 1730*x^12 - 3915*x^11 - 3705*x^10 + 2992*x^9 + 4200*x^8 - 438*x^7 - 2312*x^6 - 744*x^5 + 312*x^4 + 200*x^3 + 9*x^2 - 9*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 21 x^{19} - 3 x^{18} + 186 x^{17} + 57 x^{16} - 895 x^{15} - 438 x^{14} + 2493 x^{13} + 1730 x^{12} - 3915 x^{11} - 3705 x^{10} + 2992 x^{9} + 4200 x^{8} - 438 x^{7} - 2312 x^{6} - 744 x^{5} + 312 x^{4} + 200 x^{3} + 9 x^{2} - 9 x - 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-152136328493452883717548161594456983867=-\,7^{15}\cdot 223\cdot 80317\cdot 1789163334752592959\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.80$
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:  $7, 223, 80317, 1789163334752592959$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{7} - \frac{1}{3} a^{6} + \frac{4}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{5} - \frac{2}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{4}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{2}{9} a^{6} + \frac{1}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{9} a^{18} - \frac{1}{9} a^{13} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} - \frac{4}{9} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{2}{9} a^{4} + \frac{1}{3} a^{2} - \frac{2}{9} a + \frac{1}{3}$

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:  $15$
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:  \( 346568702169 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T159:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 384072192000
The 1165 conjugacy class representatives for t21n159 are not computed
Character table for t21n159 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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 $18{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ $21$ $18{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.9.0.1}{9} }$ $15{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $15{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
223Data not computed
80317Data not computed
1789163334752592959Data not computed