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Properties

Label 14.4.170...179.1
Degree $14$
Signature $[4, 5]$
Discriminant $-1.700\times 10^{21}$
Root discriminant \(32.84\)
Ramified primes $7,11,173$
Class number $1$
Class group trivial
Galois group $C_2\wr F_7$ (as 14T48)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 28*x^12 - 77*x^11 + 136*x^10 - 141*x^9 + 5*x^8 + 265*x^7 - 464*x^6 + 419*x^5 - 140*x^4 - 99*x^3 + 153*x^2 - 79*x + 18)
 
gp: K = bnfinit(y^14 - 7*y^13 + 28*y^12 - 77*y^11 + 136*y^10 - 141*y^9 + 5*y^8 + 265*y^7 - 464*y^6 + 419*y^5 - 140*y^4 - 99*y^3 + 153*y^2 - 79*y + 18, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 7*x^13 + 28*x^12 - 77*x^11 + 136*x^10 - 141*x^9 + 5*x^8 + 265*x^7 - 464*x^6 + 419*x^5 - 140*x^4 - 99*x^3 + 153*x^2 - 79*x + 18);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 7*x^13 + 28*x^12 - 77*x^11 + 136*x^10 - 141*x^9 + 5*x^8 + 265*x^7 - 464*x^6 + 419*x^5 - 140*x^4 - 99*x^3 + 153*x^2 - 79*x + 18)
 

\( x^{14} - 7 x^{13} + 28 x^{12} - 77 x^{11} + 136 x^{10} - 141 x^{9} + 5 x^{8} + 265 x^{7} - 464 x^{6} + \cdots + 18 \) Copy content Toggle raw display

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

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[4, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1700018408193379992179\) \(\medspace = -\,7^{8}\cdot 11\cdot 173^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(32.84\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $7^{2/3}11^{1/2}173^{1/2}\approx 159.63131361268375$
Ramified primes:   \(7\), \(11\), \(173\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-11}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4201}a^{12}-\frac{6}{4201}a^{11}+\frac{619}{4201}a^{10}+\frac{1161}{4201}a^{9}+\frac{1152}{4201}a^{8}+\frac{963}{4201}a^{7}-\frac{252}{4201}a^{6}-\frac{613}{4201}a^{5}-\frac{285}{4201}a^{4}-\frac{340}{4201}a^{3}+\frac{1616}{4201}a^{2}+\frac{185}{4201}a-\frac{1140}{4201}$, $\frac{1}{298271}a^{13}+\frac{29}{298271}a^{12}+\frac{8811}{298271}a^{11}-\frac{27586}{298271}a^{10}-\frac{80042}{298271}a^{9}+\frac{137906}{298271}a^{8}-\frac{138788}{298271}a^{7}+\frac{70386}{298271}a^{6}-\frac{13338}{298271}a^{5}+\frac{145122}{298271}a^{4}+\frac{35927}{298271}a^{3}-\frac{144903}{298271}a^{2}+\frac{131365}{298271}a-\frac{102915}{298271}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{583}{4201}a^{12}-\frac{3498}{4201}a^{11}+\frac{12194}{4201}a^{10}-\frac{28905}{4201}a^{9}+\frac{37265}{4201}a^{8}-\frac{14108}{4201}a^{7}-\frac{50293}{4201}a^{6}+\frac{117334}{4201}a^{5}-\frac{98939}{4201}a^{4}+\frac{24433}{4201}a^{3}+\frac{38913}{4201}a^{2}-\frac{34979}{4201}a+\frac{11741}{4201}$, $\frac{8}{4201}a^{12}-\frac{48}{4201}a^{11}+\frac{751}{4201}a^{10}-\frac{3315}{4201}a^{9}+\frac{9216}{4201}a^{8}-\frac{17502}{4201}a^{7}+\frac{10587}{4201}a^{6}+\frac{16101}{4201}a^{5}-\frac{48491}{4201}a^{4}+\frac{56094}{4201}a^{3}+\frac{4526}{4201}a^{2}-\frac{27927}{4201}a+\frac{11885}{4201}$, $\frac{24825}{298271}a^{13}-\frac{188023}{298271}a^{12}+\frac{775784}{298271}a^{11}-\frac{2157379}{298271}a^{10}+\frac{3878361}{298271}a^{9}-\frac{3835310}{298271}a^{8}-\frac{503673}{298271}a^{7}+\frac{8441559}{298271}a^{6}-\frac{13757068}{298271}a^{5}+\frac{9845407}{298271}a^{4}-\frac{1143764}{298271}a^{3}-\frac{4591699}{298271}a^{2}+\frac{3376456}{298271}a-\frac{1008223}{298271}$, $\frac{24825}{298271}a^{13}-\frac{134702}{298271}a^{12}+\frac{455858}{298271}a^{11}-\frac{1066677}{298271}a^{10}+\frac{1357506}{298271}a^{9}-\frac{870634}{298271}a^{8}-\frac{756433}{298271}a^{7}+\frac{2461442}{298271}a^{6}-\frac{2000462}{298271}a^{5}+\frac{2105697}{298271}a^{4}-\frac{2271457}{298271}a^{3}+\frac{3129764}{298271}a^{2}-\frac{2269251}{298271}a+\frac{842747}{298271}$, $\frac{754}{4201}a^{12}-\frac{4524}{4201}a^{11}+\frac{17219}{4201}a^{10}-\frac{44625}{4201}a^{9}+\frac{70418}{4201}a^{8}-\frac{63686}{4201}a^{7}-\frac{30370}{4201}a^{6}+\frac{176350}{4201}a^{5}-\frac{244297}{4201}a^{4}+\frac{184745}{4201}a^{3}+\frac{33782}{4201}a^{2}-\frac{95766}{4201}a+\frac{68861}{4201}$, $\frac{162}{4201}a^{12}-\frac{972}{4201}a^{11}+\frac{3655}{4201}a^{10}-\frac{9365}{4201}a^{9}+\frac{14383}{4201}a^{8}-\frac{12034}{4201}a^{7}-\frac{7216}{4201}a^{6}+\frac{35126}{4201}a^{5}-\frac{46170}{4201}a^{4}+\frac{33141}{4201}a^{3}+\frac{1330}{4201}a^{2}-\frac{12040}{4201}a+\frac{12767}{4201}$, $\frac{222891}{298271}a^{13}-\frac{1518975}{298271}a^{12}+\frac{5917010}{298271}a^{11}-\frac{15827304}{298271}a^{10}+\frac{26515749}{298271}a^{9}-\frac{24381866}{298271}a^{8}-\frac{6390975}{298271}a^{7}+\frac{60029843}{298271}a^{6}-\frac{89801716}{298271}a^{5}+\frac{68285697}{298271}a^{4}-\frac{9828863}{298271}a^{3}-\frac{29133112}{298271}a^{2}+\frac{26470479}{298271}a-\frac{9382661}{298271}$, $\frac{138109}{298271}a^{13}-\frac{813041}{298271}a^{12}+\frac{2892354}{298271}a^{11}-\frac{6974933}{298271}a^{10}+\frac{9385708}{298271}a^{9}-\frac{4850952}{298271}a^{8}-\frac{11143496}{298271}a^{7}+\frac{29444405}{298271}a^{6}-\frac{28830049}{298271}a^{5}+\frac{12213179}{298271}a^{4}+\frac{10329520}{298271}a^{3}-\frac{11099957}{298271}a^{2}+\frac{5292195}{298271}a+\frac{706585}{298271}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 1070462.31054 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 1070462.31054 \cdot 1}{2\cdot\sqrt{1700018408193379992179}}\cr\approx \mathstrut & 2.03392014477 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 28*x^12 - 77*x^11 + 136*x^10 - 141*x^9 + 5*x^8 + 265*x^7 - 464*x^6 + 419*x^5 - 140*x^4 - 99*x^3 + 153*x^2 - 79*x + 18)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^14 - 7*x^13 + 28*x^12 - 77*x^11 + 136*x^10 - 141*x^9 + 5*x^8 + 265*x^7 - 464*x^6 + 419*x^5 - 140*x^4 - 99*x^3 + 153*x^2 - 79*x + 18, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^14 - 7*x^13 + 28*x^12 - 77*x^11 + 136*x^10 - 141*x^9 + 5*x^8 + 265*x^7 - 464*x^6 + 419*x^5 - 140*x^4 - 99*x^3 + 153*x^2 - 79*x + 18);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 7*x^13 + 28*x^12 - 77*x^11 + 136*x^10 - 141*x^9 + 5*x^8 + 265*x^7 - 464*x^6 + 419*x^5 - 140*x^4 - 99*x^3 + 153*x^2 - 79*x + 18);
 
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

$C_2\wr F_7$ (as 14T48):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 5376
The 40 conjugacy class representatives for $C_2\wr F_7$
Character table for $C_2\wr F_7$

Intermediate fields

7.7.12431698517.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 14 sibling: data not computed
Degree 28 siblings: data not computed
Degree 32 sibling: 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 ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ R R ${\href{/padicField/13.14.0.1}{14} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.14.0.1}{14} }$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
\(11\) Copy content Toggle raw display 11.2.1.1$x^{2} + 22$$2$$1$$1$$C_2$$[\ ]_{2}$
11.12.0.1$x^{12} + x^{8} + x^{7} + 4 x^{6} + 2 x^{5} + 5 x^{4} + 5 x^{3} + 6 x^{2} + 5 x + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(173\) Copy content Toggle raw display 173.2.0.1$x^{2} + 169 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
173.12.6.1$x^{12} + 166080 x^{11} + 11492737040 x^{10} + 424158720576054 x^{9} + 8805540020116718882 x^{8} + 97495071516899524368908 x^{7} + 449779301463929709020590133 x^{6} + 16964380073566362435417602 x^{5} + 452675157414451740521406274 x^{4} + 12168303409330101504459930560 x^{3} + 60508335269881430680824338656 x^{2} + 48881051734527149377663539696 x + 1202696058989876603202512272$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$