Properties

Label 21.3.682...552.1
Degree $21$
Signature $[3, 9]$
Discriminant $-6.827\times 10^{24}$
Root discriminant $15.23$
Ramified primes $2, 71$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_3\times D_7$ (as 21T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - x^19 - 3*x^18 + 10*x^17 + 13*x^16 + x^15 - 27*x^14 - 16*x^13 - 20*x^12 + 55*x^11 - 4*x^10 + 8*x^9 - 32*x^8 - x^7 + 70*x^6 - 52*x^5 - 4*x^4 + 6*x^3 + 2*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^21 - 2*x^20 - x^19 - 3*x^18 + 10*x^17 + 13*x^16 + x^15 - 27*x^14 - 16*x^13 - 20*x^12 + 55*x^11 - 4*x^10 + 8*x^9 - 32*x^8 - x^7 + 70*x^6 - 52*x^5 - 4*x^4 + 6*x^3 + 2*x^2 - 6*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 2, 6, -4, -52, 70, -1, -32, 8, -4, 55, -20, -16, -27, 1, 13, 10, -3, -1, -2, 1]);
 

\( x^{21} - 2 x^{20} - x^{19} - 3 x^{18} + 10 x^{17} + 13 x^{16} + x^{15} - 27 x^{14} - 16 x^{13} - 20 x^{12} + 55 x^{11} - 4 x^{10} + 8 x^{9} - 32 x^{8} - x^{7} + 70 x^{6} - 52 x^{5} - 4 x^{4} + 6 x^{3} + 2 x^{2} - 6 x + 1 \)

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:  \(-6826740523487474380439552\)\(\medspace = -\,2^{21}\cdot 71^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.23$
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:  $2, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{1123} a^{19} + \frac{549}{1123} a^{18} - \frac{486}{1123} a^{17} + \frac{29}{1123} a^{16} + \frac{485}{1123} a^{15} - \frac{210}{1123} a^{14} - \frac{484}{1123} a^{13} + \frac{269}{1123} a^{12} - \frac{486}{1123} a^{11} - \frac{380}{1123} a^{10} - \frac{229}{1123} a^{9} + \frac{184}{1123} a^{8} + \frac{226}{1123} a^{7} - \frac{126}{1123} a^{6} - \frac{383}{1123} a^{5} - \frac{240}{1123} a^{4} + \frac{135}{1123} a^{3} - \frac{73}{1123} a^{2} + \frac{400}{1123} a - \frac{482}{1123}$, $\frac{1}{5759846840051113} a^{20} - \frac{1831290216085}{5759846840051113} a^{19} + \frac{1578844555870078}{5759846840051113} a^{18} - \frac{2737252450552301}{5759846840051113} a^{17} + \frac{427915791087797}{5759846840051113} a^{16} + \frac{2790640140238848}{5759846840051113} a^{15} + \frac{1535080883554518}{5759846840051113} a^{14} + \frac{1760669225346069}{5759846840051113} a^{13} - \frac{445066428164865}{5759846840051113} a^{12} + \frac{1330609776501890}{5759846840051113} a^{11} + \frac{962987638183864}{5759846840051113} a^{10} - \frac{690862674853154}{5759846840051113} a^{9} + \frac{1345081312925895}{5759846840051113} a^{8} + \frac{513812502314985}{5759846840051113} a^{7} - \frac{2445770844664860}{5759846840051113} a^{6} + \frac{1969042718365795}{5759846840051113} a^{5} + \frac{1833979453799909}{5759846840051113} a^{4} - \frac{2097781925797724}{5759846840051113} a^{3} + \frac{2561737255419719}{5759846840051113} a^{2} - \frac{1218171558187049}{5759846840051113} a - \frac{2486611624028086}{5759846840051113}$

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 $2$)
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:  \( 7666.61577427 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 7666.61577427 \cdot 1}{2\sqrt{6826740523487474380439552}}\approx 0.179133085640$ (assuming GRH)

Galois group

$S_3\times D_7$ (as 21T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 84
The 15 conjugacy class representatives for $S_3\times D_7$
Character table for $S_3\times D_7$

Intermediate fields

3.3.568.1, 7.1.357911.1

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 $21$ ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.7.0.1$x^{7} - x + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
2.14.21.6$x^{14} + 4 x^{11} - 3 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
71Data not computed