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Properties

Label 14.2.117244558738203125.1
Degree $14$
Signature $[2, 6]$
Discriminant $1.172\times 10^{17}$
Root discriminant \(16.57\)
Ramified primes $5,107$
Class number $1$
Class group trivial
Galois group $D_{14}$ (as 14T3)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^12 - 6*x^11 + 16*x^10 + x^9 - 3*x^8 - 34*x^7 + 37*x^6 - 10*x^5 + 14*x^4 - 7*x^3 - 20*x^2 + 16*x - 1)
 
Copy content gp:K = bnfinit(y^14 - 3*y^12 - 6*y^11 + 16*y^10 + y^9 - 3*y^8 - 34*y^7 + 37*y^6 - 10*y^5 + 14*y^4 - 7*y^3 - 20*y^2 + 16*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^12 - 6*x^11 + 16*x^10 + x^9 - 3*x^8 - 34*x^7 + 37*x^6 - 10*x^5 + 14*x^4 - 7*x^3 - 20*x^2 + 16*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^14 - 3*x^12 - 6*x^11 + 16*x^10 + x^9 - 3*x^8 - 34*x^7 + 37*x^6 - 10*x^5 + 14*x^4 - 7*x^3 - 20*x^2 + 16*x - 1)
 

\( x^{14} - 3 x^{12} - 6 x^{11} + 16 x^{10} + x^{9} - 3 x^{8} - 34 x^{7} + 37 x^{6} - 10 x^{5} + 14 x^{4} + \cdots - 1 \) 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:  $[2, 6]$
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:   \(117244558738203125\) \(\medspace = 5^{7}\cdot 107^{6}\) 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:  \(16.57\)
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:  $5^{1/2}107^{1/2}\approx 23.130067012440755$
Ramified primes:   \(5\), \(107\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{1}{9}a^{7}+\frac{2}{9}a^{6}-\frac{4}{9}a^{5}+\frac{4}{9}a^{4}-\frac{2}{9}a^{3}-\frac{4}{9}a^{2}+\frac{1}{3}a+\frac{2}{9}$, $\frac{1}{243}a^{13}-\frac{13}{243}a^{12}+\frac{4}{243}a^{11}+\frac{23}{243}a^{10}-\frac{40}{243}a^{9}+\frac{35}{243}a^{8}+\frac{28}{243}a^{7}+\frac{88}{243}a^{6}+\frac{4}{9}a^{5}+\frac{44}{243}a^{4}+\frac{10}{27}a^{3}-\frac{43}{243}a^{2}+\frac{53}{243}a+\frac{56}{243}$ 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:  No
Index:  Not computed
Inessential primes:  $3$

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:  $7$
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:   $\frac{188}{243}a^{13}+\frac{202}{243}a^{12}-\frac{409}{243}a^{11}-\frac{1670}{243}a^{10}+\frac{1255}{243}a^{9}+\frac{2044}{243}a^{8}+\frac{1484}{243}a^{7}-\frac{5299}{243}a^{6}+\frac{7}{3}a^{5}-\frac{17}{243}a^{4}+\frac{266}{27}a^{3}+\frac{1501}{243}a^{2}-\frac{2672}{243}a-\frac{218}{243}$, $\frac{14}{243}a^{13}-\frac{47}{243}a^{12}-\frac{133}{243}a^{11}-\frac{83}{243}a^{10}+\frac{601}{243}a^{9}+\frac{4}{243}a^{8}-\frac{391}{243}a^{7}-\frac{1090}{243}a^{6}+\frac{16}{3}a^{5}-\frac{221}{243}a^{4}+\frac{47}{27}a^{3}-\frac{413}{243}a^{2}-\frac{554}{243}a+\frac{406}{243}$, $\frac{32}{81}a^{13}+\frac{16}{81}a^{12}-\frac{88}{81}a^{11}-\frac{236}{81}a^{10}+\frac{394}{81}a^{9}+\frac{229}{81}a^{8}+\frac{32}{81}a^{7}-\frac{1045}{81}a^{6}+8a^{5}-\frac{131}{81}a^{4}+\frac{41}{9}a^{3}+\frac{109}{81}a^{2}-\frac{437}{81}a+\frac{118}{81}$, $\frac{151}{243}a^{13}+\frac{35}{243}a^{12}-\frac{557}{243}a^{11}-\frac{1306}{243}a^{10}+\frac{2006}{243}a^{9}+\frac{1478}{243}a^{8}+\frac{286}{243}a^{7}-\frac{5801}{243}a^{6}+\frac{92}{9}a^{5}-\frac{187}{243}a^{4}+\frac{310}{27}a^{3}+\frac{500}{243}a^{2}-\frac{3094}{243}a+\frac{464}{243}$, $\frac{56}{243}a^{13}+\frac{1}{243}a^{12}-\frac{262}{243}a^{11}-\frac{494}{243}a^{10}+\frac{1000}{243}a^{9}+\frac{988}{243}a^{8}-\frac{214}{243}a^{7}-\frac{2929}{243}a^{6}+\frac{14}{9}a^{5}+\frac{925}{243}a^{4}+\frac{146}{27}a^{3}+\frac{751}{243}a^{2}-\frac{1811}{243}a-\frac{104}{243}$, $\frac{115}{243}a^{13}+\frac{98}{243}a^{12}-\frac{215}{243}a^{11}-\frac{838}{243}a^{10}+\frac{1016}{243}a^{9}+\frac{623}{243}a^{8}+\frac{655}{243}a^{7}-\frac{2975}{243}a^{6}+\frac{71}{9}a^{5}-\frac{961}{243}a^{4}+\frac{157}{27}a^{3}+\frac{185}{243}a^{2}-\frac{1438}{243}a+\frac{878}{243}$, $\frac{109}{243}a^{13}-\frac{13}{243}a^{12}-\frac{428}{243}a^{11}-\frac{814}{243}a^{10}+\frac{1850}{243}a^{9}+\frac{737}{243}a^{8}-\frac{377}{243}a^{7}-\frac{4691}{243}a^{6}+13a^{5}-\frac{118}{243}a^{4}+\frac{238}{27}a^{3}-\frac{745}{243}a^{2}-\frac{2566}{243}a+\frac{731}{243}$ 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:  \( 1181.3184922748676 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 1181.3184922748676 \cdot 1}{2\cdot\sqrt{117244558738203125}}\cr\approx \mathstrut & 0.424550992027155 \end{aligned}\]

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 - 3*x^12 - 6*x^11 + 16*x^10 + x^9 - 3*x^8 - 34*x^7 + 37*x^6 - 10*x^5 + 14*x^4 - 7*x^3 - 20*x^2 + 16*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 - 3*x^12 - 6*x^11 + 16*x^10 + x^9 - 3*x^8 - 34*x^7 + 37*x^6 - 10*x^5 + 14*x^4 - 7*x^3 - 20*x^2 + 16*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 - 3*x^12 - 6*x^11 + 16*x^10 + x^9 - 3*x^8 - 34*x^7 + 37*x^6 - 10*x^5 + 14*x^4 - 7*x^3 - 20*x^2 + 16*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 - 3*x^12 - 6*x^11 + 16*x^10 + x^9 - 3*x^8 - 34*x^7 + 37*x^6 - 10*x^5 + 14*x^4 - 7*x^3 - 20*x^2 + 16*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

$D_{14}$ (as 14T3):

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 solvable group of order 28
The 10 conjugacy class representatives for $D_{14}$
Character table for $D_{14}$

Intermediate fields

\(\Q(\sqrt{5}) \), 7.1.153130375.1

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

Sibling fields

Galois closure: deg 28
Degree 14 sibling: 14.0.2509033556997546875.1
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 ${\href{/padicField/2.14.0.1}{14} }$ ${\href{/padicField/3.2.0.1}{2} }^{7}$ R ${\href{/padicField/7.14.0.1}{14} }$ ${\href{/padicField/11.7.0.1}{7} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{7}$ ${\href{/padicField/17.14.0.1}{14} }$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{7}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{7}$ ${\href{/padicField/41.7.0.1}{7} }^{2}$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.2.0.1}{2} }^{7}$ ${\href{/padicField/53.2.0.1}{2} }^{7}$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(107\) Copy content Toggle raw display 107.2.1.0a1.1$x^{2} + 103 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
107.2.2.2a1.2$x^{4} + 206 x^{3} + 10613 x^{2} + 412 x + 111$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
107.2.2.2a1.2$x^{4} + 206 x^{3} + 10613 x^{2} + 412 x + 111$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
107.2.2.2a1.2$x^{4} + 206 x^{3} + 10613 x^{2} + 412 x + 111$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*28 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*28 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.535.2t1.a.a$1$ $ 5 \cdot 107 $ \(\Q(\sqrt{-535}) \) $C_2$ (as 2T1) $1$ $-1$
1.107.2t1.a.a$1$ $ 107 $ \(\Q(\sqrt{-107}) \) $C_2$ (as 2T1) $1$ $-1$
*28 2.535.14t3.a.c$2$ $ 5 \cdot 107 $ 14.2.117244558738203125.1 $D_{14}$ (as 14T3) $1$ $0$
*28 2.535.7t2.a.c$2$ $ 5 \cdot 107 $ 7.1.153130375.1 $D_{7}$ (as 7T2) $1$ $0$
*28 2.535.7t2.a.b$2$ $ 5 \cdot 107 $ 7.1.153130375.1 $D_{7}$ (as 7T2) $1$ $0$
*28 2.535.7t2.a.a$2$ $ 5 \cdot 107 $ 7.1.153130375.1 $D_{7}$ (as 7T2) $1$ $0$
*28 2.535.14t3.a.b$2$ $ 5 \cdot 107 $ 14.2.117244558738203125.1 $D_{14}$ (as 14T3) $1$ $0$
*28 2.535.14t3.a.a$2$ $ 5 \cdot 107 $ 14.2.117244558738203125.1 $D_{14}$ (as 14T3) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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