Properties

Label 21.1.126...441.1
Degree $21$
Signature $[1, 10]$
Discriminant $1.261\times 10^{27}$
Root discriminant $19.52$
Ramified primes $31, 71$
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 + 3*x^19 - x^18 + x^17 + 8*x^16 - 7*x^15 + 24*x^14 - 15*x^13 + 7*x^12 + 25*x^11 - 42*x^10 + 46*x^9 - 39*x^8 - 12*x^7 + 15*x^6 - 14*x^5 + x^4 + 10*x^3 - x^2 + 2*x + 1)
 
gp: K = bnfinit(x^21 + 3*x^19 - x^18 + x^17 + 8*x^16 - 7*x^15 + 24*x^14 - 15*x^13 + 7*x^12 + 25*x^11 - 42*x^10 + 46*x^9 - 39*x^8 - 12*x^7 + 15*x^6 - 14*x^5 + x^4 + 10*x^3 - x^2 + 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -1, 10, 1, -14, 15, -12, -39, 46, -42, 25, 7, -15, 24, -7, 8, 1, -1, 3, 0, 1]);
 

\( x^{21} + 3 x^{19} - x^{18} + x^{17} + 8 x^{16} - 7 x^{15} + 24 x^{14} - 15 x^{13} + 7 x^{12} + 25 x^{11} - 42 x^{10} + 46 x^{9} - 39 x^{8} - 12 x^{7} + 15 x^{6} - 14 x^{5} + x^{4} + 10 x^{3} - x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1261412107834594448324876441\)\(\medspace = 31^{7}\cdot 71^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.52$
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:  $31, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{299} a^{19} + \frac{8}{23} a^{18} + \frac{122}{299} a^{17} - \frac{79}{299} a^{16} - \frac{41}{299} a^{15} + \frac{19}{299} a^{14} + \frac{119}{299} a^{13} - \frac{81}{299} a^{12} + \frac{132}{299} a^{11} - \frac{64}{299} a^{10} + \frac{120}{299} a^{9} + \frac{77}{299} a^{8} - \frac{4}{23} a^{7} + \frac{11}{299} a^{6} + \frac{40}{299} a^{5} + \frac{128}{299} a^{4} + \frac{131}{299} a^{3} + \frac{75}{299} a^{2} + \frac{142}{299} a + \frac{58}{299}$, $\frac{1}{31329330929} a^{20} + \frac{10588327}{31329330929} a^{19} - \frac{15216296701}{31329330929} a^{18} + \frac{10121876874}{31329330929} a^{17} - \frac{361413200}{1362144823} a^{16} + \frac{4705840582}{31329330929} a^{15} + \frac{5011990975}{31329330929} a^{14} - \frac{1631283704}{31329330929} a^{13} - \frac{9509682454}{31329330929} a^{12} + \frac{13199176568}{31329330929} a^{11} + \frac{982873072}{31329330929} a^{10} - \frac{1044585243}{2409948533} a^{9} + \frac{12174708862}{31329330929} a^{8} - \frac{13010489871}{31329330929} a^{7} - \frac{12732688830}{31329330929} a^{6} + \frac{5435101835}{31329330929} a^{5} - \frac{1553313246}{31329330929} a^{4} + \frac{6110604571}{31329330929} a^{3} - \frac{357606914}{31329330929} a^{2} - \frac{11723869589}{31329330929} a + \frac{13268663735}{31329330929}$

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:  $10$
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:  \( 80948.9311635 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{10}\cdot 80948.9311635 \cdot 1}{2\sqrt{1261412107834594448324876441}}\approx 0.218565345970$ (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

3.1.31.1, 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 $21$ ${\href{/LocalNumberField/3.14.0.1}{14} }{,}\,{\href{/LocalNumberField/3.7.0.1}{7} }$ $21$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ R ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$71$71.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$