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Properties

Label 21.5.26930966980...4016.2
Degree $21$
Signature $[5, 8]$
Discriminant $2^{18}\cdot 3^{18}\cdot 7^{36}$
Root discriminant $130.53$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T158

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-216, -756, -882, -343, 0, 0, 0, 360, 840, 490, 0, 0, 0, 0, -102, -119, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 119*x^15 - 102*x^14 + 490*x^9 + 840*x^8 + 360*x^7 - 343*x^3 - 882*x^2 - 756*x - 216)
 
gp: K = bnfinit(x^21 - 119*x^15 - 102*x^14 + 490*x^9 + 840*x^8 + 360*x^7 - 343*x^3 - 882*x^2 - 756*x - 216, 1)
 

Normalized defining polynomial

\( x^{21} - 119 x^{15} - 102 x^{14} + 490 x^{9} + 840 x^{8} + 360 x^{7} - 343 x^{3} - 882 x^{2} - 756 x - 216 \)

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:  \(269309669801808513835250991351016462192214016=2^{18}\cdot 3^{18}\cdot 7^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.53$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} - \frac{4}{13} a^{2} - \frac{5}{13} a - \frac{4}{13}$, $\frac{1}{13} a^{15} - \frac{4}{13} a^{3} - \frac{5}{13} a^{2} - \frac{4}{13} a$, $\frac{1}{13} a^{16} - \frac{4}{13} a^{4} - \frac{5}{13} a^{3} - \frac{4}{13} a^{2}$, $\frac{1}{13} a^{17} - \frac{4}{13} a^{5} - \frac{5}{13} a^{4} - \frac{4}{13} a^{3}$, $\frac{1}{13} a^{18} - \frac{4}{13} a^{6} - \frac{5}{13} a^{5} - \frac{4}{13} a^{4}$, $\frac{1}{3639168} a^{19} + \frac{6665}{303264} a^{18} + \frac{529}{33696} a^{17} + \frac{17}{4212} a^{16} - \frac{43}{2808} a^{15} - \frac{1}{78} a^{14} + \frac{132781}{279936} a^{13} + \frac{14357}{46656} a^{12} - \frac{2051}{7776} a^{11} + \frac{293}{1296} a^{10} - \frac{11}{216} a^{9} + \frac{17}{36} a^{8} + \frac{303509}{1819584} a^{7} - \frac{3}{13} a^{6} + \frac{2}{13} a^{5} + \frac{1}{13} a^{4} - \frac{6}{13} a^{3} + \frac{4}{13} a^{2} + \frac{559529}{3639168} a - \frac{233329}{606528}$, $\frac{1}{1018734133248} a^{20} + \frac{6665}{169789022208} a^{19} + \frac{44422225}{28298170368} a^{18} + \frac{31731929}{4716361728} a^{17} - \frac{17376863}{786060288} a^{16} - \frac{3909463}{131010048} a^{15} - \frac{32094055991}{1018734133248} a^{14} - \frac{3509}{11664} a^{13} + \frac{779}{1944} a^{12} - \frac{65}{324} a^{11} + \frac{17}{54} a^{10} + \frac{4}{9} a^{9} + \frac{169789022453}{509367066624} a^{8} + \frac{125615}{6530347008} a^{7} - \frac{3}{13} a^{6} - \frac{5}{13} a^{5} + \frac{5}{13} a^{4} + \frac{3}{13} a^{3} - \frac{36168075763}{78364164096} a^{2} + \frac{32650591919}{84894511104} a + \frac{8706802745}{28298170368}$

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:  \( 185340932015000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T158:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 192036096000
The 587 conjugacy class representatives for t21n158 are not computed
Character table for t21n158 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 R R $21$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $21$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $21$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.12.16$x^{12} - 16 x^{10} - 23 x^{8} + 24 x^{6} - 29 x^{4} - 8 x^{2} - 13$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.9.9.9$x^{9} + 18 x^{5} + 27 x^{2} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.7$x^{9} + 18 x^{3} + 54 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
7Data not computed