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Properties

Label 14.2.973177856947837.1
Degree $14$
Signature $[2, 6]$
Discriminant $9.732\times 10^{14}$
Root discriminant \(11.76\)
Ramified primes $71,107$
Class number $1$
Class group trivial
Galois group $C_2\wr D_7$ (as 14T38)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 3*x^12 - 13*x^10 + 27*x^9 - 4*x^8 - 31*x^7 + 36*x^6 - 14*x^5 - 13*x^4 + 22*x^3 - 15*x^2 + 6*x - 1)
 
gp: K = bnfinit(y^14 - 3*y^13 + 3*y^12 - 13*y^10 + 27*y^9 - 4*y^8 - 31*y^7 + 36*y^6 - 14*y^5 - 13*y^4 + 22*y^3 - 15*y^2 + 6*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 + 3*x^12 - 13*x^10 + 27*x^9 - 4*x^8 - 31*x^7 + 36*x^6 - 14*x^5 - 13*x^4 + 22*x^3 - 15*x^2 + 6*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^13 + 3*x^12 - 13*x^10 + 27*x^9 - 4*x^8 - 31*x^7 + 36*x^6 - 14*x^5 - 13*x^4 + 22*x^3 - 15*x^2 + 6*x - 1)
 

\( x^{14} - 3 x^{13} + 3 x^{12} - 13 x^{10} + 27 x^{9} - 4 x^{8} - 31 x^{7} + 36 x^{6} - 14 x^{5} + \cdots - 1 \) 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:  $[2, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(973177856947837\) \(\medspace = 71^{7}\cdot 107\) 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:  \(11.76\)
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:  $71^{3/4}107^{1/2}\approx 253.00881373347607$
Ramified primes:   \(71\), \(107\) 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:  \(\Q(\sqrt{7597}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11}a^{11}+\frac{2}{11}a^{10}+\frac{2}{11}a^{9}+\frac{3}{11}a^{8}-\frac{1}{11}a^{7}-\frac{3}{11}a^{6}+\frac{4}{11}a^{5}-\frac{4}{11}a^{4}+\frac{4}{11}a^{3}+\frac{4}{11}a+\frac{3}{11}$, $\frac{1}{11}a^{12}-\frac{2}{11}a^{10}-\frac{1}{11}a^{9}+\frac{4}{11}a^{8}-\frac{1}{11}a^{7}-\frac{1}{11}a^{6}-\frac{1}{11}a^{5}+\frac{1}{11}a^{4}+\frac{3}{11}a^{3}+\frac{4}{11}a^{2}-\frac{5}{11}a+\frac{5}{11}$, $\frac{1}{583}a^{13}-\frac{23}{583}a^{12}-\frac{14}{583}a^{11}-\frac{144}{583}a^{10}+\frac{58}{583}a^{9}+\frac{245}{583}a^{8}+\frac{78}{583}a^{7}-\frac{107}{583}a^{6}-\frac{156}{583}a^{5}+\frac{138}{583}a^{4}+\frac{195}{583}a^{3}-\frac{9}{583}a^{2}+\frac{6}{583}a-\frac{8}{583}$ 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:  $7$
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:   $\frac{929}{583}a^{13}-\frac{1810}{583}a^{12}+\frac{562}{583}a^{11}+\frac{1162}{583}a^{10}-\frac{10831}{583}a^{9}+\frac{13061}{583}a^{8}+\frac{13685}{583}a^{7}-\frac{17995}{583}a^{6}+\frac{687}{53}a^{5}+\frac{1956}{583}a^{4}-\frac{9645}{583}a^{3}+\frac{6903}{583}a^{2}-\frac{269}{53}a+\frac{1048}{583}$, $\frac{335}{583}a^{13}-\frac{974}{583}a^{12}+\frac{94}{53}a^{11}-\frac{116}{583}a^{10}-\frac{4420}{583}a^{9}+\frac{8988}{583}a^{8}-\frac{2066}{583}a^{7}-\frac{9027}{583}a^{6}+\frac{14043}{583}a^{5}-\frac{6664}{583}a^{4}-\frac{4688}{583}a^{3}+\frac{8168}{583}a^{2}-\frac{6417}{583}a+\frac{190}{53}$, $\frac{119}{583}a^{13}-\frac{34}{583}a^{12}-\frac{553}{583}a^{11}+\frac{672}{583}a^{10}-\frac{1154}{583}a^{9}-\frac{1002}{583}a^{8}+\frac{7215}{583}a^{7}-\frac{1868}{583}a^{6}-\frac{7487}{583}a^{5}+\frac{5928}{583}a^{4}-\frac{963}{583}a^{3}-\frac{3668}{583}a^{2}+\frac{301}{53}a-\frac{1005}{583}$, $\frac{248}{583}a^{13}-\frac{775}{583}a^{12}+\frac{65}{53}a^{11}+\frac{116}{583}a^{10}-\frac{3159}{583}a^{9}+\frac{6753}{583}a^{8}-\frac{266}{583}a^{7}-\frac{9046}{583}a^{6}+\frac{6945}{583}a^{5}-\frac{1498}{583}a^{4}-\frac{2891}{583}a^{3}+\frac{4658}{583}a^{2}-\frac{2328}{583}a+\frac{75}{53}$, $\frac{70}{53}a^{13}-\frac{1757}{583}a^{12}+\frac{827}{583}a^{11}+\frac{1109}{583}a^{10}-\frac{9294}{583}a^{9}+\frac{13273}{583}a^{8}+\frac{9763}{583}a^{7}-\frac{21228}{583}a^{6}+\frac{8299}{583}a^{5}+\frac{5242}{583}a^{4}-\frac{9751}{583}a^{3}+\frac{6161}{583}a^{2}-\frac{1899}{583}a+\frac{571}{583}$, $\frac{119}{583}a^{13}-\frac{405}{583}a^{12}+\frac{772}{583}a^{11}-\frac{600}{583}a^{10}-\frac{1631}{583}a^{9}+\frac{4404}{583}a^{8}-\frac{4816}{583}a^{7}-\frac{808}{583}a^{6}+\frac{12176}{583}a^{5}-\frac{9071}{583}a^{4}-\frac{2023}{583}a^{3}+\frac{6508}{583}a^{2}-\frac{5858}{583}a+\frac{2281}{583}$, $\frac{522}{583}a^{13}-\frac{1512}{583}a^{12}+\frac{1225}{583}a^{11}+\frac{71}{53}a^{10}-\frac{7460}{583}a^{9}+\frac{13569}{583}a^{8}+\frac{1284}{583}a^{7}-\frac{21404}{583}a^{6}+\frac{1636}{53}a^{5}+\frac{592}{583}a^{4}-\frac{11577}{583}a^{3}+\frac{11626}{583}a^{2}-\frac{4712}{583}a+\frac{435}{583}$ 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:  \( 50.753333025 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 50.753333025 \cdot 1}{2\cdot\sqrt{973177856947837}}\cr\approx \mathstrut & 0.20020626543 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 3*x^12 - 13*x^10 + 27*x^9 - 4*x^8 - 31*x^7 + 36*x^6 - 14*x^5 - 13*x^4 + 22*x^3 - 15*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^14 - 3*x^13 + 3*x^12 - 13*x^10 + 27*x^9 - 4*x^8 - 31*x^7 + 36*x^6 - 14*x^5 - 13*x^4 + 22*x^3 - 15*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^14 - 3*x^13 + 3*x^12 - 13*x^10 + 27*x^9 - 4*x^8 - 31*x^7 + 36*x^6 - 14*x^5 - 13*x^4 + 22*x^3 - 15*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^13 + 3*x^12 - 13*x^10 + 27*x^9 - 4*x^8 - 31*x^7 + 36*x^6 - 14*x^5 - 13*x^4 + 22*x^3 - 15*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

$C_2\wr D_7$ (as 14T38):

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 1792
The 40 conjugacy class representatives for $C_2\wr D_7$
Character table for $C_2\wr D_7$

Intermediate fields

7.1.357911.1

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 siblings: data not computed
Degree 28 siblings: data not computed
Degree 32 sibling: 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 ${\href{/padicField/2.14.0.1}{14} }$ ${\href{/padicField/3.7.0.1}{7} }^{2}$ ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.2.0.1}{2} }^{7}$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(71\) Copy content Toggle raw display $\Q_{71}$$x + 64$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 64$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.3.1$x^{4} + 71$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
\(107\) Copy content Toggle raw display $\Q_{107}$$x + 105$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 105$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 105$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 105$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 105$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 105$$1$$1$$0$Trivial$[\ ]$
107.2.0.1$x^{2} + 103 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} + 103 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.1.1$x^{2} + 214$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.0.1$x^{2} + 103 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$