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Properties

Label 21.9.15589694504...9641.1
Degree $21$
Signature $[9, 6]$
Discriminant $7^{14}\cdot 593^{3}\cdot 1033^{3}$
Root discriminant $24.55$
Ramified primes $7, 593, 1033$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T56

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 0, -33, 26, 171, 36, -298, -220, -39, 432, -23, -266, 375, 63, -281, 12, 93, -4, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 15*x^19 - 4*x^18 + 93*x^17 + 12*x^16 - 281*x^15 + 63*x^14 + 375*x^13 - 266*x^12 - 23*x^11 + 432*x^10 - 39*x^9 - 220*x^8 - 298*x^7 + 36*x^6 + 171*x^5 + 26*x^4 - 33*x^3 - 1)
 
gp: K = bnfinit(x^21 - 15*x^19 - 4*x^18 + 93*x^17 + 12*x^16 - 281*x^15 + 63*x^14 + 375*x^13 - 266*x^12 - 23*x^11 + 432*x^10 - 39*x^9 - 220*x^8 - 298*x^7 + 36*x^6 + 171*x^5 + 26*x^4 - 33*x^3 - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 15 x^{19} - 4 x^{18} + 93 x^{17} + 12 x^{16} - 281 x^{15} + 63 x^{14} + 375 x^{13} - 266 x^{12} - 23 x^{11} + 432 x^{10} - 39 x^{9} - 220 x^{8} - 298 x^{7} + 36 x^{6} + 171 x^{5} + 26 x^{4} - 33 x^{3} - 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(155896945048377999491545839641=7^{14}\cdot 593^{3}\cdot 1033^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.55$
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:  $7, 593, 1033$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2663940783269672738804407} a^{20} - \frac{1108193198713363173390215}{2663940783269672738804407} a^{19} - \frac{280619471151190939279528}{2663940783269672738804407} a^{18} + \frac{1209305204744802051763785}{2663940783269672738804407} a^{17} + \frac{1050844979362380785033930}{2663940783269672738804407} a^{16} - \frac{488623592641786566777228}{2663940783269672738804407} a^{15} + \frac{577355186122337007487985}{2663940783269672738804407} a^{14} + \frac{603263260087974047378388}{2663940783269672738804407} a^{13} - \frac{389489834413244620747733}{2663940783269672738804407} a^{12} + \frac{1310486505281433663334754}{2663940783269672738804407} a^{11} - \frac{1048729983896256325090096}{2663940783269672738804407} a^{10} + \frac{390217669516582945802006}{2663940783269672738804407} a^{9} + \frac{189726055948152998344830}{2663940783269672738804407} a^{8} + \frac{501274615489938950449587}{2663940783269672738804407} a^{7} + \frac{979874334769603361288035}{2663940783269672738804407} a^{6} + \frac{691130524777384509222477}{2663940783269672738804407} a^{5} + \frac{1094668764384174267729213}{2663940783269672738804407} a^{4} - \frac{412755546857005305279249}{2663940783269672738804407} a^{3} - \frac{543834043020582656504093}{2663940783269672738804407} a^{2} - \frac{209500131528907771489760}{2663940783269672738804407} a - \frac{89332762952120196100386}{2663940783269672738804407}$

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:  $14$
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:  \( 4936254.92703 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T56:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15120
The 45 conjugacy class representatives for t21n56
Character table for t21n56 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.3.612569.1

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

Sibling fields

Degree 42 sibling: 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 $21$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ R $21$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{3}$ $21$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ $21$ $15{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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

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
7Data not computed
593Data not computed
1033Data not computed