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Properties

Label 21.3.378...943.3
Degree $21$
Signature $[3, 9]$
Discriminant $-3.788\times 10^{29}$
Root discriminant \(25.62\)
Ramified prime $7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $F_7$ (as 21T4)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^21 - x^14 - 9*x^7 + 1)
 
Copy content gp:K = bnfinit(y^21 - y^14 - 9*y^7 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^14 - 9*x^7 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - x^14 - 9*x^7 + 1)
 

\( x^{21} - x^{14} - 9x^{7} + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $21$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[3, 9]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-378818692265664781682717625943\) \(\medspace = -\,7^{35}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(25.62\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $7^{71/42}\approx 26.829842007890413$
Ramified primes:   \(7\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{-7}) \)
$\Aut(K/\Q)$:   $C_3$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{14}-\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{4}a$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{5}$, $\frac{1}{4}a^{20}-\frac{1}{4}a^{6}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $11$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{1}{2}a^{19}-\frac{1}{2}a^{12}-4a^{5}$, $\frac{1}{4}a^{17}-\frac{9}{4}a^{3}$, $\frac{1}{2}a^{20}-\frac{1}{4}a^{17}-\frac{1}{2}a^{13}+\frac{1}{2}a^{10}+\frac{1}{2}a^{8}-4a^{6}+\frac{7}{4}a^{3}-a^{2}-\frac{1}{2}a-1$, $\frac{1}{2}a^{19}+\frac{1}{2}a^{18}+\frac{1}{4}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-5a^{5}-4a^{4}-a^{3}-\frac{7}{4}a-1$, $a^{20}-\frac{1}{2}a^{18}+\frac{1}{4}a^{17}+\frac{1}{4}a^{16}-\frac{1}{4}a^{15}-a^{13}+\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-9a^{6}+4a^{4}-\frac{7}{4}a^{3}-\frac{13}{4}a^{2}+\frac{9}{4}a$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-\frac{1}{4}a^{16}-\frac{1}{2}a^{12}+\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-5a^{5}+4a^{4}-4a^{3}+\frac{9}{4}a^{2}-a$, $\frac{1}{2}a^{19}-\frac{1}{4}a^{18}-\frac{1}{4}a^{17}+\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{2}a^{12}-4a^{5}+\frac{9}{4}a^{4}+\frac{9}{4}a^{3}-\frac{9}{4}a^{2}+\frac{5}{4}$, $a^{20}-\frac{1}{2}a^{18}-\frac{1}{4}a^{17}+\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-a^{13}+\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-9a^{6}+4a^{4}+\frac{9}{4}a^{3}-\frac{9}{4}a-\frac{7}{4}$, $\frac{1}{2}a^{20}-\frac{3}{4}a^{19}+\frac{1}{4}a^{18}-\frac{1}{4}a^{15}-\frac{1}{2}a^{13}+a^{12}-\frac{1}{2}a^{9}-4a^{6}+\frac{27}{4}a^{5}-\frac{9}{4}a^{4}+\frac{1}{2}a^{2}+\frac{5}{4}a-1$, $\frac{5}{4}a^{20}-\frac{5}{4}a^{19}-\frac{1}{2}a^{18}+a^{17}+\frac{1}{4}a^{16}-\frac{1}{2}a^{15}-a^{13}+a^{12}+\frac{1}{2}a^{11}-a^{10}+\frac{1}{2}a^{8}-\frac{45}{4}a^{6}+\frac{45}{4}a^{5}+5a^{4}-9a^{3}-\frac{9}{4}a^{2}+4a$, $\frac{1}{4}a^{20}+\frac{1}{4}a^{19}-\frac{1}{4}a^{18}-\frac{1}{2}a^{12}+\frac{1}{2}a^{11}-\frac{9}{4}a^{6}-\frac{11}{4}a^{5}+\frac{11}{4}a^{4}-a^{3}+a^{2}-a$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 5449377.399032222 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 5449377.399032222 \cdot 1}{2\cdot\sqrt{378818692265664781682717625943}}\cr\approx \mathstrut & 0.540517790116752 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^21 - x^14 - 9*x^7 + 1) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^21 - x^14 - 9*x^7 + 1, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - x^14 - 9*x^7 + 1); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^21 - x^14 - 9*x^7 + 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$F_7$ (as 21T4):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.1.1977326743.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 42
Degree 7 sibling: 7.1.1977326743.1
Degree 14 sibling: deg 14
Minimal sibling: 7.1.1977326743.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{7}$ ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ R ${\href{/padicField/11.3.0.1}{3} }^{7}$ ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.3.0.1}{3} }^{7}$ ${\href{/padicField/29.7.0.1}{7} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.3.0.1}{3} }^{7}$ ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.7.0.1}{7} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.3.0.1}{3} }^{7}$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.1.21.35a3.1$x^{21} + 21 x^{15} + 7$$21$$1$$35$21T4not computed

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)