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Properties

Label 14.0.9745585291264.1
Degree $14$
Signature $(0, 7)$
Discriminant $-9.746\times 10^{12}$
Root discriminant \(8.47\)
Ramified primes $2,29$
Class number $1$
Class group trivial
Galois group $C_7 \wr C_2$ (as 14T8)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 1)
 
Copy content gp:K = bnfinit(y^14 - 4*y^13 + 7*y^12 - 4*y^11 - 8*y^10 + 24*y^9 - 30*y^8 + 16*y^7 + 13*y^6 - 38*y^5 + 46*y^4 - 36*y^3 + 19*y^2 - 6*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 1)
 

\( x^{14} - 4 x^{13} + 7 x^{12} - 4 x^{11} - 8 x^{10} + 24 x^{9} - 30 x^{8} + 16 x^{7} + 13 x^{6} + \cdots + 1 \) 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:  $14$
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:  $(0, 7)$
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:   \(-9745585291264\) \(\medspace = -\,2^{14}\cdot 29^{6}\) 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:  \(8.47\)
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:  $2\cdot 29^{6/7}\approx 35.8520500359704$
Ramified primes:   \(2\), \(29\) 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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_7$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-1}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$ 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:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$
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 $1$
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:  $6$
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:   \( -a^{11} + 2 a^{10} - 2 a^{9} - 2 a^{8} + 7 a^{7} - 10 a^{6} + 4 a^{5} + 5 a^{4} - 13 a^{3} + 12 a^{2} - 8 a + 2 \)  (order $4$) 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:   $a^{12}-2a^{11}+2a^{10}+2a^{9}-7a^{8}+10a^{7}-4a^{6}-5a^{5}+13a^{4}-12a^{3}+8a^{2}-2a$, $4a^{13}-10a^{12}+10a^{11}+6a^{10}-30a^{9}+47a^{8}-29a^{7}-13a^{6}+54a^{5}-64a^{4}+51a^{3}-22a^{2}+5a+1$, $a^{13}-5a^{12}+8a^{11}-2a^{10}-14a^{9}+29a^{8}-28a^{7}+3a^{6}+29a^{5}-44a^{4}+38a^{3}-21a^{2}+5a+1$, $5a^{13}-16a^{12}+22a^{11}-3a^{10}-41a^{9}+86a^{8}-83a^{7}+18a^{6}+74a^{5}-131a^{4}+130a^{3}-83a^{2}+33a-6$, $3a^{13}-9a^{12}+10a^{11}+4a^{10}-27a^{9}+43a^{8}-29a^{7}-11a^{6}+49a^{5}-59a^{4}+46a^{3}-21a^{2}+3a+1$, $a^{12}-3a^{11}+3a^{10}+2a^{9}-9a^{8}+13a^{7}-8a^{6}-5a^{5}+16a^{4}-18a^{3}+13a^{2}-5a$ Copy content Toggle raw display
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:  \( 5.342840135225723 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 5.342840135225723 \cdot 1}{4\cdot\sqrt{9745585291264}}\cr\approx \mathstrut & 0.165412110762191 \end{aligned}\]

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^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 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^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 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^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 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^14 - 4*x^13 + 7*x^12 - 4*x^11 - 8*x^10 + 24*x^9 - 30*x^8 + 16*x^7 + 13*x^6 - 38*x^5 + 46*x^4 - 36*x^3 + 19*x^2 - 6*x + 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

$C_7\times D_7$ (as 14T8):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 98
The 35 conjugacy class representatives for $C_7 \wr C_2$
Character table for $C_7 \wr C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \)

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

Degree 14 siblings: 14.0.5796901408038404767744.4, deg 14
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.7.0.1}{7} }^{2}$ ${\href{/padicField/7.14.0.1}{14} }$ ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.7.0.1}{7} }^{2}$ ${\href{/padicField/17.7.0.1}{7} }^{2}$ ${\href{/padicField/19.14.0.1}{14} }$ ${\href{/padicField/23.14.0.1}{14} }$ R ${\href{/padicField/31.14.0.1}{14} }$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.7.0.1}{7} }^{2}$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{7}$ ${\href{/padicField/59.2.0.1}{2} }^{7}$

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
\(2\) Copy content Toggle raw display 2.7.2.14a1.1$x^{14} + 2 x^{8} + 4 x^{7} + x^{2} + 4 x + 5$$2$$7$$14$$C_{14}$$$[2]^{7}$$
\(29\) Copy content Toggle raw display 29.7.1.0a1.1$x^{7} + 2 x + 27$$1$$7$$0$$C_7$$$[\ ]^{7}$$
29.1.7.6a1.3$x^{7} + 87$$7$$1$$6$$C_7$$$[\ ]_{7}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*98 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*98 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.116.14t1.b.a$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.116.14t1.b.f$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.116.14t1.b.e$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.116.14t1.b.d$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.29.7t1.a.c$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
1.29.7t1.a.d$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
1.29.7t1.a.b$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
1.29.7t1.a.f$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
1.29.7t1.a.e$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
1.116.14t1.b.b$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.116.14t1.b.c$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.29.7t1.a.a$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
2.3364.14t8.c.e$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.7t2.a.b$2$ $ 2^{2} \cdot 29^{2}$ 7.1.38068692544.1 $D_{7}$ (as 7T2) $1$ $0$
2.3364.14t8.c.f$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.c.c$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.c.b$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.c.d$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.d.b$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.116.14t8.b.d$2$ $ 2^{2} \cdot 29 $ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.116.14t8.b.f$2$ $ 2^{2} \cdot 29 $ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.7t2.a.c$2$ $ 2^{2} \cdot 29^{2}$ 7.1.38068692544.1 $D_{7}$ (as 7T2) $1$ $0$
2.3364.14t8.d.f$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.116.14t8.b.a$2$ $ 2^{2} \cdot 29 $ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.d.e$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.d.a$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.116.14t8.b.c$2$ $ 2^{2} \cdot 29 $ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.d.c$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.d.d$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.116.14t8.b.b$2$ $ 2^{2} \cdot 29 $ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
*98 2.116.14t8.b.e$2$ $ 2^{2} \cdot 29 $ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.c.a$2$ $ 2^{2} \cdot 29^{2}$ 14.0.9745585291264.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.7t2.a.a$2$ $ 2^{2} \cdot 29^{2}$ 7.1.38068692544.1 $D_{7}$ (as 7T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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