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Properties

Label 14.4.334624313614643.1
Degree $14$
Signature $[4, 5]$
Discriminant $-3.346\times 10^{14}$
Root discriminant \(10.90\)
Ramified primes $13,83,109$
Class number $1$
Class group trivial
Galois group $C_2^7.\PSL(2,7)$ (as 14T51)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

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

\( x^{14} - x^{13} - 4 x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{9} + 2 x^{8} - 12 x^{7} + x^{5} + 2 x^{4} + \cdots - 1 \) 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:  $[4, 5]$
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:   \(-334624313614643\) \(\medspace = -\,13^{4}\cdot 83\cdot 109^{4}\) 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:  \(10.90\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $13^{1/2}83^{1/2}109^{1/2}\approx 342.94460194031336$
Ramified primes:   \(13\), \(83\), \(109\) 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:  \(\Q(\sqrt{-83}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{83}a^{13}+\frac{41}{83}a^{12}-\frac{25}{83}a^{11}+\frac{32}{83}a^{10}+\frac{19}{83}a^{9}-\frac{29}{83}a^{8}+\frac{29}{83}a^{7}-\frac{39}{83}a^{6}+\frac{22}{83}a^{5}+\frac{12}{83}a^{4}+\frac{8}{83}a^{3}+\frac{10}{83}a^{2}+\frac{2}{83}a+\frac{2}{83}$ 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:  Not computed
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:  $8$
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:   $\frac{8}{83}a^{13}-\frac{4}{83}a^{12}-\frac{34}{83}a^{11}+\frac{7}{83}a^{10}+\frac{69}{83}a^{9}+\frac{17}{83}a^{8}-\frac{100}{83}a^{7}-\frac{63}{83}a^{6}+\frac{10}{83}a^{5}+\frac{179}{83}a^{4}+\frac{147}{83}a^{3}-\frac{169}{83}a^{2}-\frac{67}{83}a-\frac{67}{83}$, $a$, $\frac{36}{83}a^{13}-\frac{18}{83}a^{12}-\frac{153}{83}a^{11}+\frac{73}{83}a^{10}+\frac{103}{83}a^{9}+\frac{35}{83}a^{8}+\frac{214}{83}a^{7}-\frac{325}{83}a^{6}-\frac{121}{83}a^{5}+\frac{100}{83}a^{4}-\frac{127}{83}a^{3}+\frac{277}{83}a^{2}-\frac{11}{83}a-\frac{11}{83}$, $\frac{12}{83}a^{13}-\frac{6}{83}a^{12}-\frac{51}{83}a^{11}+\frac{52}{83}a^{10}+\frac{62}{83}a^{9}-\frac{99}{83}a^{8}-\frac{67}{83}a^{7}-\frac{53}{83}a^{6}+\frac{181}{83}a^{5}+\frac{227}{83}a^{4}-\frac{70}{83}a^{3}-\frac{46}{83}a^{2}-\frac{142}{83}a+\frac{24}{83}$, $\frac{5}{83}a^{13}+\frac{39}{83}a^{12}-\frac{42}{83}a^{11}-\frac{172}{83}a^{10}+\frac{95}{83}a^{9}+\frac{187}{83}a^{8}+\frac{62}{83}a^{7}+\frac{54}{83}a^{6}-\frac{388}{83}a^{5}-\frac{106}{83}a^{4}+\frac{206}{83}a^{3}+\frac{50}{83}a^{2}+\frac{176}{83}a+\frac{10}{83}$, $\frac{39}{83}a^{13}-\frac{61}{83}a^{12}-\frac{145}{83}a^{11}+\frac{252}{83}a^{10}+\frac{77}{83}a^{9}-\frac{135}{83}a^{8}+\frac{52}{83}a^{7}-\frac{442}{83}a^{6}+\frac{277}{83}a^{5}+\frac{302}{83}a^{4}-\frac{186}{83}a^{3}+\frac{224}{83}a^{2}-\frac{254}{83}a-\frac{5}{83}$, $\frac{3}{83}a^{13}-\frac{43}{83}a^{12}+\frac{8}{83}a^{11}+\frac{179}{83}a^{10}-\frac{26}{83}a^{9}-\frac{170}{83}a^{8}-\frac{162}{83}a^{7}-\frac{117}{83}a^{6}+\frac{398}{83}a^{5}+\frac{202}{83}a^{4}-\frac{59}{83}a^{3}-\frac{53}{83}a^{2}-\frac{243}{83}a+\frac{6}{83}$, $\frac{8}{83}a^{13}-\frac{4}{83}a^{12}-\frac{34}{83}a^{11}+\frac{7}{83}a^{10}+\frac{69}{83}a^{9}+\frac{17}{83}a^{8}-\frac{100}{83}a^{7}-\frac{63}{83}a^{6}+\frac{10}{83}a^{5}+\frac{96}{83}a^{4}+\frac{147}{83}a^{3}-\frac{3}{83}a^{2}-\frac{67}{83}a-\frac{67}{83}$ 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:  \( 37.8345982696 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 37.8345982696 \cdot 1}{2\cdot\sqrt{334624313614643}}\cr\approx \mathstrut & 0.162031595726 \end{aligned}\]

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 - x^13 - 4*x^12 + 3*x^11 + 3*x^10 + 3*x^9 + 2*x^8 - 12*x^7 + x^5 + 2*x^4 + 6*x^3 - 3*x^2 + 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 - x^13 - 4*x^12 + 3*x^11 + 3*x^10 + 3*x^9 + 2*x^8 - 12*x^7 + x^5 + 2*x^4 + 6*x^3 - 3*x^2 + 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 - x^13 - 4*x^12 + 3*x^11 + 3*x^10 + 3*x^9 + 2*x^8 - 12*x^7 + x^5 + 2*x^4 + 6*x^3 - 3*x^2 + 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 = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 - 4*x^12 + 3*x^11 + 3*x^10 + 3*x^9 + 2*x^8 - 12*x^7 + x^5 + 2*x^4 + 6*x^3 - 3*x^2 + 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

$C_2^7.\PSL(2,7)$ (as 14T51):

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); G, transitive_group_identification(G)
 
A non-solvable group of order 21504
The 48 conjugacy class representatives for $C_2^7.\PSL(2,7)$
Character table for $C_2^7.\PSL(2,7)$

Intermediate fields

7.3.2007889.2

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 14 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 siblings: data not computed
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 ${\href{/padicField/2.14.0.1}{14} }$ ${\href{/padicField/3.7.0.1}{7} }^{2}$ ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ R ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.7.0.1}{7} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(13\) Copy content Toggle raw display 13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.4.2.4a1.2$x^{8} + 6 x^{6} + 24 x^{5} + 13 x^{4} + 72 x^{3} + 156 x^{2} + 48 x + 17$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(83\) Copy content Toggle raw display 83.2.1.0a1.1$x^{2} + 82 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
83.1.2.1a1.1$x^{2} + 83$$2$$1$$1$$C_2$$$[\ ]_{2}$$
83.2.1.0a1.1$x^{2} + 82 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
83.4.1.0a1.1$x^{4} + 4 x^{2} + 42 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
83.4.1.0a1.1$x^{4} + 4 x^{2} + 42 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(109\) Copy content Toggle raw display $\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
109.4.2.4a1.2$x^{8} + 22 x^{6} + 196 x^{5} + 133 x^{4} + 2156 x^{3} + 9736 x^{2} + 1176 x + 145$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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