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Properties

Label 21.21.2137431244...0000.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{20}\cdot 3^{9}\cdot 5^{9}\cdot 7^{21}\cdot 37^{7}$
Root discriminant $144.07$
Ramified primes $2, 3, 5, 7, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![686, 2744, -10976, -35672, 54880, 148176, -117992, -290570, 120344, 304976, -58604, -178752, 12838, 59584, -1008, -11256, 0, 1148, 0, -56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 56*x^19 + 1148*x^17 - 11256*x^15 - 1008*x^14 + 59584*x^13 + 12838*x^12 - 178752*x^11 - 58604*x^10 + 304976*x^9 + 120344*x^8 - 290570*x^7 - 117992*x^6 + 148176*x^5 + 54880*x^4 - 35672*x^3 - 10976*x^2 + 2744*x + 686)
 
gp: K = bnfinit(x^21 - 56*x^19 + 1148*x^17 - 11256*x^15 - 1008*x^14 + 59584*x^13 + 12838*x^12 - 178752*x^11 - 58604*x^10 + 304976*x^9 + 120344*x^8 - 290570*x^7 - 117992*x^6 + 148176*x^5 + 54880*x^4 - 35672*x^3 - 10976*x^2 + 2744*x + 686, 1)
 

Normalized defining polynomial

\( x^{21} - 56 x^{19} + 1148 x^{17} - 11256 x^{15} - 1008 x^{14} + 59584 x^{13} + 12838 x^{12} - 178752 x^{11} - 58604 x^{10} + 304976 x^{9} + 120344 x^{8} - 290570 x^{7} - 117992 x^{6} + 148176 x^{5} + 54880 x^{4} - 35672 x^{3} - 10976 x^{2} + 2744 x + 686 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2137431244633054651040398357581563904000000000=2^{20}\cdot 3^{9}\cdot 5^{9}\cdot 7^{21}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.07$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{7} a^{9}$, $\frac{1}{7} a^{10}$, $\frac{1}{7} a^{11}$, $\frac{1}{7} a^{12}$, $\frac{1}{7} a^{13}$, $\frac{1}{49} a^{14}$, $\frac{1}{147} a^{15} + \frac{1}{147} a^{14} + \frac{1}{21} a^{12} + \frac{1}{21} a^{9} + \frac{1}{21} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{147} a^{16} - \frac{1}{147} a^{14} + \frac{1}{21} a^{13} - \frac{1}{21} a^{12} + \frac{1}{21} a^{10} - \frac{1}{21} a^{9} + \frac{1}{21} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{147} a^{17} - \frac{1}{147} a^{14} - \frac{1}{21} a^{13} + \frac{1}{21} a^{12} + \frac{1}{21} a^{11} - \frac{1}{21} a^{10} - \frac{1}{21} a^{9} - \frac{1}{21} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{147} a^{18} + \frac{1}{21} a^{13} - \frac{1}{21} a^{12} - \frac{1}{21} a^{11} - \frac{1}{21} a^{10} + \frac{1}{21} a^{9} - \frac{1}{21} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{147} a^{19} + \frac{1}{147} a^{14} - \frac{1}{21} a^{13} - \frac{1}{21} a^{12} - \frac{1}{21} a^{11} + \frac{1}{21} a^{10} - \frac{1}{21} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{68751991904143191} a^{20} - \frac{194761020394828}{68751991904143191} a^{19} + \frac{106812334923265}{68751991904143191} a^{18} + \frac{14073621845373}{22917330634714397} a^{17} - \frac{71473948254131}{22917330634714397} a^{16} + \frac{115630267879781}{68751991904143191} a^{15} + \frac{76320076046117}{9821713129163313} a^{14} + \frac{344020403999479}{9821713129163313} a^{13} + \frac{176318287003400}{3273904376387771} a^{12} + \frac{260550801612148}{9821713129163313} a^{11} + \frac{230917470076740}{3273904376387771} a^{10} + \frac{866675821813}{16127607765457} a^{9} + \frac{43520718486983}{1403101875594759} a^{8} - \frac{420029848244960}{9821713129163313} a^{7} - \frac{199612005640549}{467700625198253} a^{6} + \frac{165638726333230}{467700625198253} a^{5} + \frac{277555254311101}{1403101875594759} a^{4} + \frac{214290721245082}{467700625198253} a^{3} - \frac{202574861267426}{467700625198253} a^{2} - \frac{110614200289445}{467700625198253} a - \frac{399233346565340}{1403101875594759}$

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $20$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23839453211300000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.3.148.1, 7.7.177885288000.1

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 R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R $21$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
37Data not computed