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Properties

Label 21.7.24136659873...8839.1
Degree $21$
Signature $[7, 7]$
Discriminant $-\,23^{7}\cdot 577^{9}$
Root discriminant $43.38$
Ramified primes $23, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times D_7$ (as 21T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 21, 8, -146, 11, 967, -310, -3155, 3853, 2894, -3557, -2134, 1278, 883, -306, -166, 67, 30, -7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 2*x^20 - 7*x^19 + 30*x^18 + 67*x^17 - 166*x^16 - 306*x^15 + 883*x^14 + 1278*x^13 - 2134*x^12 - 3557*x^11 + 2894*x^10 + 3853*x^9 - 3155*x^8 - 310*x^7 + 967*x^6 + 11*x^5 - 146*x^4 + 8*x^3 + 21*x^2 - 1)
 
gp: K = bnfinit(x^21 - 2*x^20 - 7*x^19 + 30*x^18 + 67*x^17 - 166*x^16 - 306*x^15 + 883*x^14 + 1278*x^13 - 2134*x^12 - 3557*x^11 + 2894*x^10 + 3853*x^9 - 3155*x^8 - 310*x^7 + 967*x^6 + 11*x^5 - 146*x^4 + 8*x^3 + 21*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 2 x^{20} - 7 x^{19} + 30 x^{18} + 67 x^{17} - 166 x^{16} - 306 x^{15} + 883 x^{14} + 1278 x^{13} - 2134 x^{12} - 3557 x^{11} + 2894 x^{10} + 3853 x^{9} - 3155 x^{8} - 310 x^{7} + 967 x^{6} + 11 x^{5} - 146 x^{4} + 8 x^{3} + 21 x^{2} - 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-24136659873181133143007749754988839=-\,23^{7}\cdot 577^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.38$
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:  $23, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{14} - \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{14} - \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{19} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{281603514833761420416820225} a^{20} - \frac{1668847576547003873174879}{56320702966752284083364045} a^{19} - \frac{14857900906774869726974512}{281603514833761420416820225} a^{18} + \frac{3917254333317361977470971}{281603514833761420416820225} a^{17} - \frac{182219487133645942044476}{281603514833761420416820225} a^{16} - \frac{19116057542599175194873268}{281603514833761420416820225} a^{15} + \frac{31972420138587674379701503}{281603514833761420416820225} a^{14} + \frac{7144213400465018991433654}{281603514833761420416820225} a^{13} + \frac{106471185536252889074596201}{281603514833761420416820225} a^{12} - \frac{65408469431171240892752652}{281603514833761420416820225} a^{11} + \frac{17341137295947223848653849}{281603514833761420416820225} a^{10} + \frac{123851879544324943751207977}{281603514833761420416820225} a^{9} + \frac{53618811940851037412866227}{281603514833761420416820225} a^{8} + \frac{107605044464510061470920734}{281603514833761420416820225} a^{7} - \frac{118255390160771186187370047}{281603514833761420416820225} a^{6} + \frac{74094334123859040408889888}{281603514833761420416820225} a^{5} - \frac{123896382612972283944791778}{281603514833761420416820225} a^{4} + \frac{99054581144029644277794748}{281603514833761420416820225} a^{3} - \frac{84383408499905824587496026}{281603514833761420416820225} a^{2} - \frac{97339186329492730213268451}{281603514833761420416820225} a + \frac{25930321623006607911271638}{281603514833761420416820225}$

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:  $13$
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:  \( 2208285999.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times D_7$ (as 21T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

3.1.23.1, 7.7.192100033.1

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 $21$ $21$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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
$23$23.7.0.1$x^{7} - x + 8$$1$$7$$0$$C_7$$[\ ]^{7}$
23.14.7.2$x^{14} - 148035889 x^{2} + 27238603576$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
577Data not computed