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Properties

Label 21.3.429...912.1
Degree $21$
Signature $[3, 9]$
Discriminant $-4.297\times 10^{27}$
Root discriminant $20.70$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $SO(3,7)$ (as 21T20)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 + 9*x^19 - 7*x^18 + 35*x^17 - 27*x^16 + 83*x^15 - 61*x^14 + 123*x^13 - 111*x^12 + 163*x^11 - 225*x^10 + 213*x^9 - 253*x^8 + 177*x^7 - 167*x^6 + 192*x^5 - 180*x^4 + 136*x^3 - 76*x^2 - 12*x + 4)
 
gp: K = bnfinit(x^21 - x^20 + 9*x^19 - 7*x^18 + 35*x^17 - 27*x^16 + 83*x^15 - 61*x^14 + 123*x^13 - 111*x^12 + 163*x^11 - 225*x^10 + 213*x^9 - 253*x^8 + 177*x^7 - 167*x^6 + 192*x^5 - 180*x^4 + 136*x^3 - 76*x^2 - 12*x + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -12, -76, 136, -180, 192, -167, 177, -253, 213, -225, 163, -111, 123, -61, 83, -27, 35, -7, 9, -1, 1]);
 

\( x^{21} - x^{20} + 9 x^{19} - 7 x^{18} + 35 x^{17} - 27 x^{16} + 83 x^{15} - 61 x^{14} + 123 x^{13} - 111 x^{12} + 163 x^{11} - 225 x^{10} + 213 x^{9} - 253 x^{8} + 177 x^{7} - 167 x^{6} + 192 x^{5} - 180 x^{4} + 136 x^{3} - 76 x^{2} - 12 x + 4 \)

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:  \(-4297390112987454978404646912\)\(\medspace = -\,2^{38}\cdot 3^{18}\cdot 7^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.70$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{12} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{13} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{14} - \frac{1}{8} a^{10} + \frac{1}{8} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{3064} a^{19} + \frac{21}{766} a^{18} + \frac{151}{3064} a^{17} - \frac{121}{3064} a^{16} + \frac{45}{1532} a^{15} + \frac{291}{3064} a^{14} + \frac{69}{1532} a^{13} + \frac{43}{766} a^{12} - \frac{375}{3064} a^{11} - \frac{21}{766} a^{10} + \frac{617}{3064} a^{9} - \frac{707}{3064} a^{8} + \frac{21}{1532} a^{7} - \frac{1287}{3064} a^{6} - \frac{109}{1532} a^{5} + \frac{175}{383} a^{4} + \frac{77}{1532} a^{3} - \frac{43}{766} a^{2} - \frac{174}{383} a - \frac{116}{383}$, $\frac{1}{641043952} a^{20} - \frac{4635}{320521976} a^{19} + \frac{32363831}{641043952} a^{18} + \frac{8251891}{320521976} a^{17} + \frac{1206609}{641043952} a^{16} + \frac{12188533}{320521976} a^{15} - \frac{75715165}{641043952} a^{14} + \frac{30270037}{320521976} a^{13} - \frac{11344357}{641043952} a^{12} - \frac{28886933}{320521976} a^{11} - \frac{33953279}{641043952} a^{10} - \frac{13960699}{320521976} a^{9} + \frac{109565635}{641043952} a^{8} + \frac{67665735}{320521976} a^{7} + \frac{49630213}{641043952} a^{6} + \frac{12984823}{320521976} a^{5} + \frac{14117977}{160260988} a^{4} + \frac{39150567}{80130494} a^{3} + \frac{17535969}{40065247} a^{2} + \frac{17297159}{80130494} a + \frac{76042019}{160260988}$

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:  \( 1790438.21987 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{9}\cdot 1790438.21987 \cdot 1}{2\sqrt{4297390112987454978404646912}}\approx 1.66738448630$ (assuming GRH)

Galois group

$SO(3,7)$ (as 21T20):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 336
The 9 conjugacy class representatives for $SO(3,7)$
Character table for $SO(3,7)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 8 sibling: data not computed
Degree 14 sibling: data not computed
Degree 16 sibling: data not computed
Degree 24 sibling: data not computed
Degree 28 siblings: 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.10.2$x^{6} + 2 x^{5} + 2 x^{4} + 2 x^{2} + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.26.65$x^{12} + 4 x^{3} + 2$$12$$1$$26$$S_4$$[8/3, 8/3]_{3}^{2}$
3Data not computed
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$