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Properties

Label 21.5.73255911656...4656.1
Degree $21$
Signature $[5, 8]$
Discriminant $2^{18}\cdot 3^{35}\cdot 7^{21}$
Root discriminant $79.13$
Ramified primes $2, 3, 7$
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![-216, -756, -882, -343, 0, 0, 0, -864, -2016, -1176, 0, 0, 0, 0, -18, -21, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^15 - 18*x^14 - 1176*x^9 - 2016*x^8 - 864*x^7 - 343*x^3 - 882*x^2 - 756*x - 216)
 
gp: K = bnfinit(x^21 - 21*x^15 - 18*x^14 - 1176*x^9 - 2016*x^8 - 864*x^7 - 343*x^3 - 882*x^2 - 756*x - 216, 1)
 

Normalized defining polynomial

\( x^{21} - 21 x^{15} - 18 x^{14} - 1176 x^{9} - 2016 x^{8} - 864 x^{7} - 343 x^{3} - 882 x^{2} - 756 x - 216 \)

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:  \(7325591165663336053947172202947791814656=2^{18}\cdot 3^{35}\cdot 7^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{7} - \frac{1}{9} a^{5} + \frac{1}{3} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{7} + \frac{1}{3} a^{6} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{13} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{2}{27} a^{8} - \frac{2}{27} a^{7} + \frac{4}{9} a^{6} + \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{27} a^{2} + \frac{10}{27} a - \frac{4}{9}$, $\frac{1}{27} a^{15} - \frac{1}{27} a^{13} + \frac{1}{9} a^{10} - \frac{2}{27} a^{9} + \frac{2}{27} a^{7} - \frac{1}{9} a^{6} - \frac{1}{9} a^{4} + \frac{10}{27} a^{3} - \frac{10}{27} a + \frac{1}{9}$, $\frac{1}{27} a^{16} + \frac{1}{27} a^{13} + \frac{1}{27} a^{10} + \frac{1}{9} a^{9} + \frac{1}{27} a^{7} + \frac{4}{9} a^{6} + \frac{1}{3} a^{5} + \frac{7}{27} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{7}{27} a + \frac{2}{9}$, $\frac{1}{27} a^{17} - \frac{1}{27} a^{13} + \frac{1}{27} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{27} a^{7} - \frac{1}{9} a^{6} - \frac{2}{27} a^{5} + \frac{1}{3} a^{4} - \frac{4}{9} a^{3} + \frac{2}{9} a^{2} - \frac{13}{27} a + \frac{4}{9}$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{13} + \frac{1}{27} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{4}{27} a^{7} + \frac{10}{27} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{2}{9} a^{2} + \frac{13}{27} a - \frac{4}{9}$, $\frac{1}{839808} a^{19} - \frac{1111}{69984} a^{18} - \frac{47}{7776} a^{17} + \frac{17}{972} a^{16} + \frac{5}{648} a^{15} - \frac{1}{54} a^{14} - \frac{5191}{279936} a^{13} + \frac{1}{46656} a^{12} + \frac{41}{7776} a^{11} - \frac{191}{1296} a^{10} + \frac{11}{72} a^{9} - \frac{17}{108} a^{8} - \frac{697}{34992} a^{7} + \frac{2}{9} a^{6} + \frac{7}{27} a^{5} - \frac{1}{9} a^{4} - \frac{13}{27} a^{3} + \frac{4}{27} a^{2} - \frac{342487}{839808} a - \frac{62257}{139968}$, $\frac{1}{705277476864} a^{20} + \frac{53321}{117546246144} a^{19} - \frac{59247407}{19591041024} a^{18} - \frac{15157351}{3265173504} a^{17} + \frac{5951137}{544195584} a^{16} + \frac{802793}{90699264} a^{15} + \frac{2362675385}{235092492288} a^{14} - \frac{115039}{2519424} a^{13} - \frac{5783}{419904} a^{12} - \frac{655}{69984} a^{11} + \frac{1081}{11664} a^{10} + \frac{113}{1944} a^{9} + \frac{1179090383}{29386561536} a^{8} + \frac{632282105}{4897760256} a^{7} - \frac{13}{27} a^{6} - \frac{1}{27} a^{5} - \frac{8}{27} a^{4} - \frac{4}{27} a^{3} + \frac{52242775721}{705277476864} a^{2} + \frac{15228331727}{58773123072} a - \frac{4356177415}{19591041024}$

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:  \( 2604814902030 \) (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

\(\Q(\zeta_{9})^+\)

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $18{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ $15{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ $18{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ $21$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ $21$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.12.12.11$x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.9.18.21$x^{9} + 6 x^{6} + 18 x^{2} + 9 x + 3$$9$$1$$18$$C_3 \wr C_3 $$[2, 2, 7/3]^{3}$
7Data not computed