Properties

Label 21.5.279...288.1
Degree $21$
Signature $[5, 8]$
Discriminant $2.798\times 10^{28}$
Root discriminant $22.63$
Ramified primes $2, 181$
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 - 2*x^19 - 7*x^18 - 5*x^17 + 11*x^16 + 17*x^15 + 34*x^14 - 32*x^13 - 19*x^12 - 120*x^11 - 65*x^10 - 276*x^9 + 43*x^8 + 539*x^7 + 983*x^6 + 606*x^5 + 280*x^4 + 121*x^3 + 40*x^2 - 13*x - 5)
 
gp: K = bnfinit(x^21 - 2*x^19 - 7*x^18 - 5*x^17 + 11*x^16 + 17*x^15 + 34*x^14 - 32*x^13 - 19*x^12 - 120*x^11 - 65*x^10 - 276*x^9 + 43*x^8 + 539*x^7 + 983*x^6 + 606*x^5 + 280*x^4 + 121*x^3 + 40*x^2 - 13*x - 5, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -13, 40, 121, 280, 606, 983, 539, 43, -276, -65, -120, -19, -32, 34, 17, 11, -5, -7, -2, 0, 1]);
 

\( x^{21} - 2 x^{19} - 7 x^{18} - 5 x^{17} + 11 x^{16} + 17 x^{15} + 34 x^{14} - 32 x^{13} - 19 x^{12} - 120 x^{11} - 65 x^{10} - 276 x^{9} + 43 x^{8} + 539 x^{7} + 983 x^{6} + 606 x^{5} + 280 x^{4} + 121 x^{3} + 40 x^{2} - 13 x - 5 \)

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[5, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(27984468103350046268523020288\)\(\medspace = 2^{27}\cdot 181^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $22.63$
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:  $2, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{16} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{17} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{41763884954038300545706964} a^{20} - \frac{4384520612991629367960817}{41763884954038300545706964} a^{19} + \frac{4919205512487759043065141}{41763884954038300545706964} a^{18} - \frac{8192924729290694549770089}{41763884954038300545706964} a^{17} - \frac{1837940363600787241765291}{20881942477019150272853482} a^{16} - \frac{1966412728332095824280412}{10440971238509575136426741} a^{15} + \frac{8631661602338128650069315}{41763884954038300545706964} a^{14} + \frac{2499023731154116546066495}{41763884954038300545706964} a^{13} - \frac{3733286767353775395163441}{41763884954038300545706964} a^{12} - \frac{18512464666113843262853227}{41763884954038300545706964} a^{11} - \frac{3768356408772525744283547}{41763884954038300545706964} a^{10} + \frac{11090519300950814557410773}{41763884954038300545706964} a^{9} + \frac{13876935540853754607682151}{41763884954038300545706964} a^{8} + \frac{4393567017349300449088311}{41763884954038300545706964} a^{7} - \frac{5057277625950989943556127}{10440971238509575136426741} a^{6} + \frac{135206788839969911531608}{10440971238509575136426741} a^{5} + \frac{171999369495583932187267}{10440971238509575136426741} a^{4} + \frac{2108802572647516743353177}{10440971238509575136426741} a^{3} + \frac{3736557581458921507577913}{41763884954038300545706964} a^{2} + \frac{7357465744758727505651809}{41763884954038300545706964} a + \frac{4998362784246496082720258}{10440971238509575136426741}$

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:  $12$
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:  \( 2296254.50823 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{8}\cdot 2296254.50823 \cdot 1}{2\sqrt{27984468103350046268523020288}}\approx 0.533482801470$ (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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
181Data not computed