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Properties

Label 14.0.104...027.1
Degree $14$
Signature $[0, 7]$
Discriminant $-1.041\times 10^{20}$
Root discriminant \(26.90\)
Ramified primes $3,547$
Class number $4$
Class group [2, 2]
Galois group $A_7\times C_2$ (as 14T47)

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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 + 10*x^12 - 19*x^11 + 22*x^10 - 46*x^9 + 23*x^8 + 223*x^7 - 394*x^6 - 152*x^5 + 734*x^4 - 191*x^3 - 473*x^2 + 185*x + 169)
 
Copy content gp:K = bnfinit(y^14 - y^13 + 10*y^12 - 19*y^11 + 22*y^10 - 46*y^9 + 23*y^8 + 223*y^7 - 394*y^6 - 152*y^5 + 734*y^4 - 191*y^3 - 473*y^2 + 185*y + 169, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 + 10*x^12 - 19*x^11 + 22*x^10 - 46*x^9 + 23*x^8 + 223*x^7 - 394*x^6 - 152*x^5 + 734*x^4 - 191*x^3 - 473*x^2 + 185*x + 169);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - x^13 + 10*x^12 - 19*x^11 + 22*x^10 - 46*x^9 + 23*x^8 + 223*x^7 - 394*x^6 - 152*x^5 + 734*x^4 - 191*x^3 - 473*x^2 + 185*x + 169)
 

\( x^{14} - x^{13} + 10 x^{12} - 19 x^{11} + 22 x^{10} - 46 x^{9} + 23 x^{8} + 223 x^{7} - 394 x^{6} + \cdots + 169 \) 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:   \(-104052649942678734027\) \(\medspace = -\,3^{19}\cdot 547^{4}\) 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:  \(26.90\)
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:  $3^{11/6}547^{2/3}\approx 501.24168878521635$
Ramified primes:   \(3\), \(547\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_2$
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 a CM field.
Reflex fields:  unavailable$^{64}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{93\cdots 87}a^{13}-\frac{33\cdots 86}{93\cdots 87}a^{12}+\frac{32\cdots 59}{93\cdots 87}a^{11}-\frac{37\cdots 20}{93\cdots 87}a^{10}-\frac{19\cdots 41}{93\cdots 87}a^{9}-\frac{36\cdots 45}{93\cdots 87}a^{8}+\frac{42\cdots 10}{93\cdots 87}a^{7}+\frac{20\cdots 19}{93\cdots 87}a^{6}-\frac{77\cdots 13}{93\cdots 87}a^{5}+\frac{30\cdots 32}{93\cdots 87}a^{4}+\frac{59\cdots 16}{93\cdots 87}a^{3}+\frac{63\cdots 20}{93\cdots 87}a^{2}-\frac{39\cdots 24}{93\cdots 87}a-\frac{85\cdots 04}{93\cdots 87}$ 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:  $C_{2}\times C_{2}$, which has order $4$
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:  $C_{2}\times C_{2}$, which has order $4$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   $4$

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:   \( \frac{870432031}{77272065779} a^{13} - \frac{1268974872}{77272065779} a^{12} + \frac{9176108091}{77272065779} a^{11} - \frac{21441193099}{77272065779} a^{10} + \frac{27490593384}{77272065779} a^{9} - \frac{59261715996}{77272065779} a^{8} + \frac{48536021240}{77272065779} a^{7} + \frac{164093826959}{77272065779} a^{6} - \frac{389886449088}{77272065779} a^{5} + \frac{34544061775}{77272065779} a^{4} + \frac{525770854031}{77272065779} a^{3} - \frac{340335303933}{77272065779} a^{2} - \frac{123765750524}{77272065779} a + \frac{177549653888}{77272065779} \)  (order $6$) 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:   $\frac{10\cdots 32}{93\cdots 87}a^{13}+\frac{178837329511485}{93\cdots 87}a^{12}+\frac{11\cdots 82}{93\cdots 87}a^{11}-\frac{68\cdots 05}{93\cdots 87}a^{10}+\frac{25\cdots 49}{93\cdots 87}a^{9}-\frac{35\cdots 31}{93\cdots 87}a^{8}+\frac{97\cdots 39}{93\cdots 87}a^{7}+\frac{17\cdots 46}{93\cdots 87}a^{6}-\frac{15\cdots 78}{93\cdots 87}a^{5}-\frac{19\cdots 74}{93\cdots 87}a^{4}+\frac{28\cdots 33}{93\cdots 87}a^{3}+\frac{44\cdots 27}{93\cdots 87}a^{2}-\frac{15\cdots 65}{93\cdots 87}a-\frac{34\cdots 06}{93\cdots 87}$, $\frac{676593162687722}{93\cdots 87}a^{13}-\frac{12\cdots 91}{93\cdots 87}a^{12}+\frac{67\cdots 16}{93\cdots 87}a^{11}-\frac{19\cdots 52}{93\cdots 87}a^{10}+\frac{19\cdots 44}{93\cdots 87}a^{9}-\frac{48\cdots 61}{93\cdots 87}a^{8}+\frac{41\cdots 02}{93\cdots 87}a^{7}+\frac{12\cdots 81}{93\cdots 87}a^{6}-\frac{36\cdots 43}{93\cdots 87}a^{5}-\frac{56\cdots 34}{93\cdots 87}a^{4}+\frac{54\cdots 46}{93\cdots 87}a^{3}-\frac{21\cdots 76}{93\cdots 87}a^{2}-\frac{24\cdots 79}{93\cdots 87}a+\frac{41\cdots 84}{93\cdots 87}$, $\frac{15\cdots 46}{93\cdots 87}a^{13}-\frac{15\cdots 52}{93\cdots 87}a^{12}+\frac{14\cdots 84}{93\cdots 87}a^{11}-\frac{31\cdots 38}{93\cdots 87}a^{10}+\frac{24\cdots 35}{93\cdots 87}a^{9}-\frac{79\cdots 80}{93\cdots 87}a^{8}+\frac{37\cdots 81}{93\cdots 87}a^{7}+\frac{36\cdots 23}{93\cdots 87}a^{6}-\frac{53\cdots 30}{93\cdots 87}a^{5}-\frac{35\cdots 13}{93\cdots 87}a^{4}+\frac{97\cdots 02}{93\cdots 87}a^{3}+\frac{99\cdots 17}{93\cdots 87}a^{2}-\frac{50\cdots 19}{93\cdots 87}a-\frac{16\cdots 35}{93\cdots 87}$, $\frac{602625997943263}{93\cdots 87}a^{13}-\frac{972935692432281}{93\cdots 87}a^{12}+\frac{60\cdots 07}{93\cdots 87}a^{11}-\frac{16\cdots 14}{93\cdots 87}a^{10}+\frac{15\cdots 38}{93\cdots 87}a^{9}-\frac{42\cdots 76}{93\cdots 87}a^{8}+\frac{22\cdots 60}{93\cdots 87}a^{7}+\frac{12\cdots 38}{93\cdots 87}a^{6}-\frac{31\cdots 26}{93\cdots 87}a^{5}+\frac{47\cdots 49}{93\cdots 87}a^{4}+\frac{35\cdots 74}{93\cdots 87}a^{3}-\frac{21\cdots 72}{93\cdots 87}a^{2}-\frac{14\cdots 33}{93\cdots 87}a+\frac{31\cdots 76}{93\cdots 87}$, $\frac{723222165407791}{93\cdots 87}a^{13}-\frac{12\cdots 78}{93\cdots 87}a^{12}+\frac{79\cdots 31}{93\cdots 87}a^{11}-\frac{19\cdots 04}{93\cdots 87}a^{10}+\frac{27\cdots 75}{93\cdots 87}a^{9}-\frac{45\cdots 55}{93\cdots 87}a^{8}+\frac{36\cdots 36}{93\cdots 87}a^{7}+\frac{17\cdots 43}{93\cdots 87}a^{6}-\frac{47\cdots 26}{93\cdots 87}a^{5}+\frac{27\cdots 97}{93\cdots 87}a^{4}+\frac{34\cdots 32}{93\cdots 87}a^{3}-\frac{40\cdots 97}{93\cdots 87}a^{2}-\frac{26\cdots 28}{93\cdots 87}a+\frac{13\cdots 82}{93\cdots 87}$, $\frac{576614519474769}{93\cdots 87}a^{13}-\frac{849769415629303}{93\cdots 87}a^{12}+\frac{67\cdots 49}{93\cdots 87}a^{11}-\frac{12\cdots 23}{93\cdots 87}a^{10}+\frac{24\cdots 12}{93\cdots 87}a^{9}-\frac{32\cdots 41}{93\cdots 87}a^{8}+\frac{22\cdots 32}{93\cdots 87}a^{7}+\frac{10\cdots 78}{93\cdots 87}a^{6}-\frac{35\cdots 99}{93\cdots 87}a^{5}+\frac{18\cdots 49}{93\cdots 87}a^{4}+\frac{47\cdots 58}{93\cdots 87}a^{3}-\frac{53\cdots 63}{93\cdots 87}a^{2}-\frac{84\cdots 37}{93\cdots 87}a+\frac{26\cdots 17}{93\cdots 87}$ 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:  \( 18139.3317602 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 18139.3317602 \cdot 4}{6\cdot\sqrt{104052649942678734027}}\cr\approx \mathstrut & 0.458313374355 \end{aligned}\]

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 + 10*x^12 - 19*x^11 + 22*x^10 - 46*x^9 + 23*x^8 + 223*x^7 - 394*x^6 - 152*x^5 + 734*x^4 - 191*x^3 - 473*x^2 + 185*x + 169) 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 + 10*x^12 - 19*x^11 + 22*x^10 - 46*x^9 + 23*x^8 + 223*x^7 - 394*x^6 - 152*x^5 + 734*x^4 - 191*x^3 - 473*x^2 + 185*x + 169, 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 + 10*x^12 - 19*x^11 + 22*x^10 - 46*x^9 + 23*x^8 + 223*x^7 - 394*x^6 - 152*x^5 + 734*x^4 - 191*x^3 - 473*x^2 + 185*x + 169); 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 - x^13 + 10*x^12 - 19*x^11 + 22*x^10 - 46*x^9 + 23*x^8 + 223*x^7 - 394*x^6 - 152*x^5 + 734*x^4 - 191*x^3 - 473*x^2 + 185*x + 169); 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

$C_2\times A_7$ (as 14T47):

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 non-solvable group of order 5040
The 18 conjugacy class representatives for $A_7\times C_2$
Character table for $A_7\times C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 7.7.1963110249.1

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 30 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 ${\href{/padicField/2.14.0.1}{14} }$ R ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.14.0.1}{14} }$

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
\(3\) Copy content Toggle raw display 3.1.2.1a1.1$x^{2} + 3$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.2.6.18a5.5$x^{12} + 18 x^{11} + 138 x^{10} + 643 x^{9} + 2064 x^{8} + 4848 x^{7} + 8576 x^{6} + 11544 x^{5} + 11760 x^{4} + 8864 x^{3} + 4704 x^{2} + 1584 x + 259$$6$$2$$18$12T40$$[2, 2]_{2}^{4}$$
\(547\) Copy content Toggle raw display $\Q_{547}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{547}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$3$$1$$2$
Deg $3$$3$$1$$2$

Spectrum of ring of integers

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