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Properties

Label 14.2.103...568.1
Degree $14$
Signature $[2, 6]$
Discriminant $1.030\times 10^{19}$
Root discriminant \(22.81\)
Ramified primes $2,3,11,13$
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 - 3*x^13 - 4*x^12 + 13*x^11 + 21*x^10 - 39*x^9 - 53*x^8 + 47*x^7 + 95*x^6 - 25*x^5 - 71*x^4 - 22*x^3 + 26*x^2 + 7*x + 1)
 
gp: K = bnfinit(y^14 - 3*y^13 - 4*y^12 + 13*y^11 + 21*y^10 - 39*y^9 - 53*y^8 + 47*y^7 + 95*y^6 - 25*y^5 - 71*y^4 - 22*y^3 + 26*y^2 + 7*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 - 4*x^12 + 13*x^11 + 21*x^10 - 39*x^9 - 53*x^8 + 47*x^7 + 95*x^6 - 25*x^5 - 71*x^4 - 22*x^3 + 26*x^2 + 7*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^13 - 4*x^12 + 13*x^11 + 21*x^10 - 39*x^9 - 53*x^8 + 47*x^7 + 95*x^6 - 25*x^5 - 71*x^4 - 22*x^3 + 26*x^2 + 7*x + 1)
 

\( x^{14} - 3 x^{13} - 4 x^{12} + 13 x^{11} + 21 x^{10} - 39 x^{9} - 53 x^{8} + 47 x^{7} + 95 x^{6} + \cdots + 1 \) 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:   \(10300792064452028568\) \(\medspace = 2^{3}\cdot 3^{4}\cdot 11^{7}\cdot 13^{8}\) 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:  \(22.81\)
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:  $2^{3/2}3^{1/2}11^{1/2}13^{2/3}\approx 89.83195783382237$
Ramified primes:   \(2\), \(3\), \(11\), \(13\) 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:  \(\Q(\sqrt{22}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{6557}a^{13}-\frac{2119}{6557}a^{12}-\frac{1188}{6557}a^{11}+\frac{30}{79}a^{10}+\frac{3009}{6557}a^{9}-\frac{236}{6557}a^{8}+\frac{991}{6557}a^{7}+\frac{1331}{6557}a^{6}+\frac{3209}{6557}a^{5}+\frac{2783}{6557}a^{4}-\frac{713}{6557}a^{3}+\frac{576}{6557}a^{2}+\frac{812}{6557}a-\frac{251}{6557}$ 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:   $\frac{1363}{6557}a^{13}-\frac{3117}{6557}a^{12}-\frac{6222}{6557}a^{11}+\frac{126}{79}a^{10}+\frac{29370}{6557}a^{9}-\frac{26603}{6557}a^{8}-\frac{59022}{6557}a^{7}+\frac{17535}{6557}a^{6}+\frac{79032}{6557}a^{5}-\frac{9831}{6557}a^{4}-\frac{34168}{6557}a^{3}-\frac{8309}{6557}a^{2}+\frac{18294}{6557}a-\frac{1149}{6557}$, $a$, $\frac{1363}{6557}a^{13}-\frac{3117}{6557}a^{12}-\frac{6222}{6557}a^{11}+\frac{126}{79}a^{10}+\frac{29370}{6557}a^{9}-\frac{26603}{6557}a^{8}-\frac{59022}{6557}a^{7}+\frac{17535}{6557}a^{6}+\frac{79032}{6557}a^{5}-\frac{9831}{6557}a^{4}-\frac{34168}{6557}a^{3}-\frac{1752}{6557}a^{2}+\frac{11737}{6557}a-\frac{1149}{6557}$, $\frac{465}{6557}a^{13}-\frac{1785}{6557}a^{12}-\frac{1632}{6557}a^{11}+\frac{125}{79}a^{10}+\frac{9101}{6557}a^{9}-\frac{37613}{6557}a^{8}-\frac{30960}{6557}a^{7}+\frac{68127}{6557}a^{6}+\frac{75873}{6557}a^{5}-\frac{63204}{6557}a^{4}-\frac{82379}{6557}a^{3}+\frac{5560}{6557}a^{2}+\frac{36616}{6557}a+\frac{1311}{6557}$, $\frac{2224}{6557}a^{13}-\frac{4730}{6557}a^{12}-\frac{12755}{6557}a^{11}+\frac{202}{79}a^{10}+\frac{62889}{6557}a^{9}-\frac{33089}{6557}a^{8}-\frac{149979}{6557}a^{7}-\frac{23291}{6557}a^{6}+\frac{212624}{6557}a^{5}+\frac{111053}{6557}a^{4}-\frac{90716}{6557}a^{3}-\frac{122174}{6557}a^{2}-\frac{16958}{6557}a-\frac{879}{6557}$, $\frac{2893}{6557}a^{13}-\frac{6029}{6557}a^{12}-\frac{14130}{6557}a^{11}+\frac{206}{79}a^{10}+\frac{62911}{6557}a^{9}-\frac{27048}{6557}a^{8}-\frac{116472}{6557}a^{7}-\frac{31161}{6557}a^{6}+\frac{110394}{6557}a^{5}+\frac{58236}{6557}a^{4}+\frac{2746}{6557}a^{3}-\frac{18781}{6557}a^{2}+\frac{1710}{6557}a-\frac{11430}{6557}$, $\frac{3419}{6557}a^{13}-\frac{12490}{6557}a^{12}-\frac{9546}{6557}a^{11}+\frac{660}{79}a^{10}+\frac{58851}{6557}a^{9}-\frac{170855}{6557}a^{8}-\frac{152551}{6557}a^{7}+\frac{229626}{6557}a^{6}+\frac{296775}{6557}a^{5}-\frac{169612}{6557}a^{4}-\frac{234595}{6557}a^{3}-\frac{17427}{6557}a^{2}+\frac{107529}{6557}a+\frac{7355}{6557}$ 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:  \( 22855.5172753 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 22855.5172753 \cdot 1}{2\cdot\sqrt{10300792064452028568}}\cr\approx \mathstrut & 0.876324493405 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 - 4*x^12 + 13*x^11 + 21*x^10 - 39*x^9 - 53*x^8 + 47*x^7 + 95*x^6 - 25*x^5 - 71*x^4 - 22*x^3 + 26*x^2 + 7*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 - 4*x^12 + 13*x^11 + 21*x^10 - 39*x^9 - 53*x^8 + 47*x^7 + 95*x^6 - 25*x^5 - 71*x^4 - 22*x^3 + 26*x^2 + 7*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 - 4*x^12 + 13*x^11 + 21*x^10 - 39*x^9 - 53*x^8 + 47*x^7 + 95*x^6 - 25*x^5 - 71*x^4 - 22*x^3 + 26*x^2 + 7*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 - 4*x^12 + 13*x^11 + 21*x^10 - 39*x^9 - 53*x^8 + 47*x^7 + 95*x^6 - 25*x^5 - 71*x^4 - 22*x^3 + 26*x^2 + 7*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

$C_2\wr F_7$ (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

7.1.38014691.1

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: 14.0.936435642222911688.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ R R ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.14.0.1}{14} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.14.0.1}{14} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(2\) Copy content Toggle raw display 2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.12.0.1$x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.1$x^{6} + 18 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.0.1$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
\(11\) Copy content Toggle raw display 11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(13\) Copy content Toggle raw display 13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.12.8.1$x^{12} + 9 x^{10} + 88 x^{9} + 33 x^{8} + 216 x^{7} - 1299 x^{6} - 78 x^{5} - 1797 x^{4} - 15494 x^{3} + 21687 x^{2} - 41586 x + 201846$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$