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Properties

Label 21.5.162...041.1
Degree $21$
Signature $[5, 8]$
Discriminant $1.625\times 10^{25}$
Root discriminant $15.87$
Ramified primes $13, 109$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\PSL(2,7)$ (as 21T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 6*x^19 + 8*x^18 + 15*x^17 - 28*x^16 - 39*x^15 + 74*x^14 + 112*x^13 - 127*x^12 - 227*x^11 + 89*x^10 + 306*x^9 + x^8 - 244*x^7 - 55*x^6 + 122*x^5 + 40*x^4 - 41*x^3 - 6*x^2 + 8*x - 1)
 
gp: K = bnfinit(x^21 - x^20 - 6*x^19 + 8*x^18 + 15*x^17 - 28*x^16 - 39*x^15 + 74*x^14 + 112*x^13 - 127*x^12 - 227*x^11 + 89*x^10 + 306*x^9 + x^8 - 244*x^7 - 55*x^6 + 122*x^5 + 40*x^4 - 41*x^3 - 6*x^2 + 8*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 8, -6, -41, 40, 122, -55, -244, 1, 306, 89, -227, -127, 112, 74, -39, -28, 15, 8, -6, -1, 1]);
 

\( x^{21} - x^{20} - 6 x^{19} + 8 x^{18} + 15 x^{17} - 28 x^{16} - 39 x^{15} + 74 x^{14} + 112 x^{13} - 127 x^{12} - 227 x^{11} + 89 x^{10} + 306 x^{9} + x^{8} - 244 x^{7} - 55 x^{6} + 122 x^{5} + 40 x^{4} - 41 x^{3} - 6 x^{2} + 8 x - 1 \)

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[5, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(16253945603436050603615041\)\(\medspace = 13^{8}\cdot 109^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.87$
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:  $13, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{13} a^{17} + \frac{3}{13} a^{16} - \frac{1}{13} a^{15} + \frac{2}{13} a^{14} - \frac{3}{13} a^{13} - \frac{5}{13} a^{12} - \frac{1}{13} a^{11} - \frac{2}{13} a^{10} + \frac{6}{13} a^{8} + \frac{5}{13} a^{7} - \frac{5}{13} a^{5} + \frac{4}{13} a^{4} - \frac{6}{13} a^{3} + \frac{6}{13} a^{2} - \frac{1}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{18} + \frac{3}{13} a^{16} + \frac{5}{13} a^{15} + \frac{4}{13} a^{14} + \frac{4}{13} a^{13} + \frac{1}{13} a^{12} + \frac{1}{13} a^{11} + \frac{6}{13} a^{10} + \frac{6}{13} a^{9} - \frac{2}{13} a^{7} - \frac{5}{13} a^{6} + \frac{6}{13} a^{5} - \frac{5}{13} a^{4} - \frac{2}{13} a^{3} - \frac{6}{13} a^{2} + \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{19} - \frac{4}{13} a^{16} - \frac{6}{13} a^{15} - \frac{2}{13} a^{14} - \frac{3}{13} a^{13} + \frac{3}{13} a^{12} - \frac{4}{13} a^{11} - \frac{1}{13} a^{10} + \frac{6}{13} a^{8} + \frac{6}{13} a^{7} + \frac{6}{13} a^{6} - \frac{3}{13} a^{5} - \frac{1}{13} a^{4} - \frac{1}{13} a^{3} - \frac{1}{13} a^{2} - \frac{3}{13}$, $\frac{1}{54746195699} a^{20} - \frac{1527135170}{54746195699} a^{19} - \frac{1923090249}{54746195699} a^{18} + \frac{600428357}{54746195699} a^{17} + \frac{3509160196}{54746195699} a^{16} - \frac{1994377962}{4211245823} a^{15} + \frac{1969036218}{54746195699} a^{14} - \frac{16609214244}{54746195699} a^{13} - \frac{4648215350}{54746195699} a^{12} + \frac{20830433798}{54746195699} a^{11} - \frac{26314124665}{54746195699} a^{10} - \frac{23928899390}{54746195699} a^{9} + \frac{21131991303}{54746195699} a^{8} - \frac{20467156920}{54746195699} a^{7} - \frac{26186883091}{54746195699} a^{6} + \frac{4504342602}{54746195699} a^{5} - \frac{1048801418}{54746195699} a^{4} - \frac{1285799224}{54746195699} a^{3} + \frac{19159610627}{54746195699} a^{2} - \frac{9682737660}{54746195699} a + \frac{7063160961}{54746195699}$

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:  \( 17240.5464964 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{8}\cdot 17240.5464964 \cdot 1}{2\sqrt{16253945603436050603615041}}\approx 0.166199822890$ (assuming GRH)

Galois group

$\PSL(2,7)$ (as 21T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 168
The 6 conjugacy class representatives for $\PSL(2,7)$
Character table for $\PSL(2,7)$

Intermediate fields

7.3.2007889.1, 7.3.2007889.2

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

Sibling fields

Degree 7 siblings: 7.3.2007889.1, 7.3.2007889.2
Degree 8 sibling: 8.0.4031618236321.1
Degree 14 siblings: Deg 14, Deg 14
Degree 24 sibling: data not computed
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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
109Data not computed