Properties

Label 21.3.837...752.1
Degree $21$
Signature $[3, 9]$
Discriminant $-8.376\times 10^{32}$
Root discriminant $36.96$
Ramified primes $2, 3, 13, 31, 47, 89$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 21T62

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 13*x^19 - 17*x^18 + 21*x^17 - 45*x^16 + 82*x^15 - 15*x^14 + 21*x^13 - 84*x^12 + 93*x^11 + 37*x^10 - 137*x^9 - 497*x^8 - 444*x^7 - 317*x^6 - 346*x^5 - 656*x^4 - 808*x^3 - 640*x^2 - 288*x - 64)
 
gp: K = bnfinit(x^21 - 6*x^20 + 13*x^19 - 17*x^18 + 21*x^17 - 45*x^16 + 82*x^15 - 15*x^14 + 21*x^13 - 84*x^12 + 93*x^11 + 37*x^10 - 137*x^9 - 497*x^8 - 444*x^7 - 317*x^6 - 346*x^5 - 656*x^4 - 808*x^3 - 640*x^2 - 288*x - 64, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, -288, -640, -808, -656, -346, -317, -444, -497, -137, 37, 93, -84, 21, -15, 82, -45, 21, -17, 13, -6, 1]);
 

\(x^{21} - 6 x^{20} + 13 x^{19} - 17 x^{18} + 21 x^{17} - 45 x^{16} + 82 x^{15} - 15 x^{14} + 21 x^{13} - 84 x^{12} + 93 x^{11} + 37 x^{10} - 137 x^{9} - 497 x^{8} - 444 x^{7} - 317 x^{6} - 346 x^{5} - 656 x^{4} - 808 x^{3} - 640 x^{2} - 288 x - 64\)  Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-837604276918124655203492919754752\)\(\medspace = -\,2^{14}\cdot 3^{7}\cdot 13^{3}\cdot 31^{3}\cdot 47^{7}\cdot 89^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $36.96$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 13, 31, 47, 89$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{9} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{3}{16} a^{8} - \frac{3}{16} a^{7} - \frac{1}{2} a^{6} - \frac{1}{16} a^{5} + \frac{3}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{10} + \frac{3}{16} a^{9} + \frac{1}{8} a^{8} - \frac{3}{16} a^{7} - \frac{3}{16} a^{6} - \frac{1}{8} a^{5} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{16} - \frac{1}{32} a^{14} + \frac{1}{32} a^{13} - \frac{1}{32} a^{11} - \frac{3}{32} a^{10} - \frac{3}{16} a^{9} + \frac{5}{32} a^{8} + \frac{7}{32} a^{7} + \frac{1}{16} a^{6} - \frac{9}{32} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{32} a^{17} - \frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{16} a^{13} - \frac{1}{32} a^{12} + \frac{3}{32} a^{11} - \frac{3}{32} a^{9} - \frac{3}{32} a^{8} - \frac{3}{8} a^{7} - \frac{1}{32} a^{6} - \frac{1}{16} a^{5} - \frac{5}{16} a^{4}$, $\frac{1}{7232} a^{18} + \frac{67}{7232} a^{17} - \frac{73}{7232} a^{16} - \frac{99}{3616} a^{15} + \frac{55}{7232} a^{14} - \frac{25}{7232} a^{13} + \frac{43}{3616} a^{12} - \frac{763}{7232} a^{11} - \frac{405}{7232} a^{10} + \frac{129}{3616} a^{9} - \frac{109}{7232} a^{8} + \frac{1925}{7232} a^{7} - \frac{1819}{7232} a^{6} - \frac{24}{113} a^{5} - \frac{235}{904} a^{4} - \frac{149}{452} a^{3} + \frac{85}{452} a^{2} - \frac{20}{113} a + \frac{23}{113}$, $\frac{1}{28928} a^{19} + \frac{23}{3616} a^{17} + \frac{399}{28928} a^{16} + \frac{439}{28928} a^{15} - \frac{47}{14464} a^{14} - \frac{725}{28928} a^{13} - \frac{423}{28928} a^{12} - \frac{655}{7232} a^{11} - \frac{857}{28928} a^{10} - \frac{445}{28928} a^{9} - \frac{1535}{7232} a^{8} + \frac{121}{452} a^{7} + \frac{11179}{28928} a^{6} - \frac{3837}{14464} a^{5} - \frac{65}{904} a^{4} + \frac{1039}{3616} a^{3} + \frac{223}{452} a^{2} - \frac{325}{904} a - \frac{185}{452}$, $\frac{1}{115712} a^{20} - \frac{1}{115712} a^{19} - \frac{1}{14464} a^{18} + \frac{7}{115712} a^{17} + \frac{175}{14464} a^{16} + \frac{3131}{115712} a^{15} + \frac{3273}{115712} a^{14} - \frac{161}{57856} a^{13} - \frac{6053}{115712} a^{12} - \frac{10845}{115712} a^{11} + \frac{1011}{28928} a^{10} + \frac{22513}{115712} a^{9} + \frac{3279}{28928} a^{8} - \frac{9989}{115712} a^{7} + \frac{50155}{115712} a^{6} - \frac{18795}{57856} a^{5} - \frac{6917}{14464} a^{4} + \frac{5981}{14464} a^{3} + \frac{1239}{3616} a^{2} + \frac{855}{3616} a - \frac{163}{1808}$  Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 679470568.354 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 679470568.354 \cdot 1}{2\sqrt{837604276918124655203492919754752}}\approx 1.43327660878$ (assuming GRH)

Galois group

21T62:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 16464
The 65 conjugacy class representatives for t21n62 are not computed
Character table for t21n62 is not computed

Intermediate fields

3.3.564.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 28 sibling: 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 ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ $21$ $21$ R ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.3.0.1}{3} }^{7}$ $21$ R ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{9}$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{7}$ R ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.7.0.1$x^{7} + x^{2} - x + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.14.0.1$x^{14} - x + 12$$1$$14$$0$$C_{14}$$[\ ]^{14}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.14.7.1$x^{14} - 207646 x^{8} + 10779215329 x^{2} - 24824532902687$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.1$x^{2} - 89$$2$$1$$1$$C_2$$[\ ]_{2}$
89.14.0.1$x^{14} - x + 15$$1$$14$$0$$C_{14}$$[\ ]^{14}$