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Properties

Label 14.0.280155320935227.1
Degree $14$
Signature $[0, 7]$
Discriminant $-2.802\times 10^{14}$
Root discriminant \(10.76\)
Ramified primes $3,71$
Class number $1$
Class group trivial
Galois group $D_{14}$ (as 14T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 2*x^12 - x^11 + 3*x^10 - 4*x^9 + 3*x^8 - 4*x^7 + 2*x^6 - 4*x^5 + 2*x^4 + 5*x^2 + 2*x + 1)
 
gp: K = bnfinit(y^14 - y^13 + 2*y^12 - y^11 + 3*y^10 - 4*y^9 + 3*y^8 - 4*y^7 + 2*y^6 - 4*y^5 + 2*y^4 + 5*y^2 + 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 + 2*x^12 - x^11 + 3*x^10 - 4*x^9 + 3*x^8 - 4*x^7 + 2*x^6 - 4*x^5 + 2*x^4 + 5*x^2 + 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 + 2*x^12 - x^11 + 3*x^10 - 4*x^9 + 3*x^8 - 4*x^7 + 2*x^6 - 4*x^5 + 2*x^4 + 5*x^2 + 2*x + 1)
 

\( x^{14} - x^{13} + 2 x^{12} - x^{11} + 3 x^{10} - 4 x^{9} + 3 x^{8} - 4 x^{7} + 2 x^{6} - 4 x^{5} + \cdots + 1 \) 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:  $[0, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-280155320935227\) \(\medspace = -\,3^{7}\cdot 71^{6}\) 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:  \(10.76\)
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:  $3^{1/2}71^{1/2}\approx 14.594519519326424$
Ramified primes:   \(3\), \(71\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{3}{7}a^{11}+\frac{2}{7}a^{10}-\frac{1}{7}a^{9}+\frac{3}{7}a^{8}-\frac{1}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}-\frac{1}{7}a^{2}+\frac{3}{7}$, $\frac{1}{119}a^{13}+\frac{2}{119}a^{12}-\frac{43}{119}a^{11}-\frac{45}{119}a^{10}+\frac{4}{119}a^{9}+\frac{59}{119}a^{8}+\frac{27}{119}a^{7}+\frac{9}{119}a^{6}+\frac{12}{119}a^{5}+\frac{32}{119}a^{4}-\frac{38}{119}a^{3}+\frac{22}{119}a^{2}+\frac{3}{119}a+\frac{11}{119}$ 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:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{47}{119} a^{13} + \frac{6}{17} a^{12} - \frac{10}{17} a^{11} + \frac{1}{17} a^{10} - \frac{86}{119} a^{9} + \frac{134}{119} a^{8} - \frac{96}{119} a^{7} + \frac{138}{119} a^{6} - \frac{88}{119} a^{5} + \frac{128}{119} a^{4} - \frac{12}{17} a^{3} + \frac{20}{119} a^{2} - \frac{260}{119} a + \frac{10}{119} \)  (order $6$) 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:   $\frac{73}{119}a^{13}-\frac{18}{17}a^{12}+\frac{30}{17}a^{11}-\frac{20}{17}a^{10}+\frac{207}{119}a^{9}-\frac{317}{119}a^{8}+\frac{339}{119}a^{7}-\frac{346}{119}a^{6}+\frac{162}{119}a^{5}-\frac{214}{119}a^{4}+\frac{19}{17}a^{3}-\frac{26}{119}a^{2}+\frac{219}{119}a-\frac{13}{119}$, $a$, $\frac{43}{119}a^{13}-\frac{12}{17}a^{12}+\frac{20}{17}a^{11}-\frac{19}{17}a^{10}+\frac{223}{119}a^{9}-\frac{353}{119}a^{8}+\frac{379}{119}a^{7}-\frac{344}{119}a^{6}+\frac{278}{119}a^{5}-\frac{307}{119}a^{4}+\frac{41}{17}a^{3}-\frac{74}{119}a^{2}+\frac{248}{119}a-\frac{37}{119}$, $\frac{73}{119}a^{13}-\frac{18}{17}a^{12}+\frac{30}{17}a^{11}-\frac{20}{17}a^{10}+\frac{207}{119}a^{9}-\frac{317}{119}a^{8}+\frac{339}{119}a^{7}-\frac{346}{119}a^{6}+\frac{162}{119}a^{5}-\frac{214}{119}a^{4}+\frac{19}{17}a^{3}-\frac{26}{119}a^{2}+\frac{219}{119}a+\frac{106}{119}$, $\frac{10}{17}a^{13}-\frac{14}{17}a^{12}+\frac{29}{17}a^{11}-\frac{25}{17}a^{10}+\frac{40}{17}a^{9}-\frac{56}{17}a^{8}+\frac{49}{17}a^{7}-\frac{63}{17}a^{6}+\frac{35}{17}a^{5}-\frac{37}{17}a^{4}+\frac{28}{17}a^{3}-\frac{1}{17}a^{2}+\frac{47}{17}a+\frac{8}{17}$, $\frac{57}{119}a^{13}-\frac{39}{119}a^{12}+\frac{65}{119}a^{11}-\frac{15}{119}a^{10}+\frac{143}{119}a^{9}-\frac{190}{119}a^{8}+\frac{26}{119}a^{7}-\frac{19}{17}a^{6}+\frac{89}{119}a^{5}-\frac{250}{119}a^{4}+\frac{27}{119}a^{3}+\frac{14}{17}a^{2}+\frac{290}{119}a+\frac{7}{17}$ 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:  \( 64.9438034848 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 64.9438034848 \cdot 1}{6\cdot\sqrt{280155320935227}}\cr\approx \mathstrut & 0.250003512392 \end{aligned}\]

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

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 28
The 10 conjugacy class representatives for $D_{14}$
Character table for $D_{14}$

Intermediate fields

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

Galois closure: deg 28
Degree 14 sibling: 14.2.19891027786401117.1
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 ${\href{/padicField/2.14.0.1}{14} }$ R ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.2.0.1}{2} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{7}$ ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{7}$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{7}$ ${\href{/padicField/29.14.0.1}{14} }$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{7}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{7}$ ${\href{/padicField/53.2.0.1}{2} }^{7}$ ${\href{/padicField/59.2.0.1}{2} }^{7}$

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
\(3\) Copy content Toggle raw display 3.14.7.2$x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
\(71\) Copy content Toggle raw display 71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$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}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.71.2t1.a.a$1$ $ 71 $ \(\Q(\sqrt{-71}) \) $C_2$ (as 2T1) $1$ $-1$
1.213.2t1.a.a$1$ $ 3 \cdot 71 $ \(\Q(\sqrt{213}) \) $C_2$ (as 2T1) $1$ $1$
* 2.71.7t2.a.c$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.a.a$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.639.14t3.a.a$2$ $ 3^{2} \cdot 71 $ 14.0.280155320935227.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.b$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.639.14t3.a.b$2$ $ 3^{2} \cdot 71 $ 14.0.280155320935227.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.639.14t3.a.c$2$ $ 3^{2} \cdot 71 $ 14.0.280155320935227.1 $D_{14}$ (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 *.