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Properties

Label 14.0.9095120158391.1
Degree $14$
Signature $[0, 7]$
Discriminant $-9.095\times 10^{12}$
Root discriminant \(8.43\)
Ramified prime $71$
Class number $1$
Class group trivial
Galois group $D_{7}$ (as 14T2)

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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 + 25*x^12 - 59*x^11 + 103*x^10 - 141*x^9 + 159*x^8 - 153*x^7 + 129*x^6 - 95*x^5 + 58*x^4 - 27*x^3 + 10*x^2 - 3*x + 1)
 
gp: K = bnfinit(y^14 - 7*y^13 + 25*y^12 - 59*y^11 + 103*y^10 - 141*y^9 + 159*y^8 - 153*y^7 + 129*y^6 - 95*y^5 + 58*y^4 - 27*y^3 + 10*y^2 - 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 7*x^13 + 25*x^12 - 59*x^11 + 103*x^10 - 141*x^9 + 159*x^8 - 153*x^7 + 129*x^6 - 95*x^5 + 58*x^4 - 27*x^3 + 10*x^2 - 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 7*x^13 + 25*x^12 - 59*x^11 + 103*x^10 - 141*x^9 + 159*x^8 - 153*x^7 + 129*x^6 - 95*x^5 + 58*x^4 - 27*x^3 + 10*x^2 - 3*x + 1)
 

\( x^{14} - 7 x^{13} + 25 x^{12} - 59 x^{11} + 103 x^{10} - 141 x^{9} + 159 x^{8} - 153 x^{7} + 129 x^{6} + \cdots + 1 \) 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:  $[0, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-9095120158391\) \(\medspace = -\,71^{7}\) 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:  \(8.43\)
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:  $71^{1/2}\approx 8.426149773176359$
Ramified primes:   \(71\) 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:  \(\Q(\sqrt{-71}) \)
$\card{ \Gal(K/\Q) }$:  $14$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{1}{7}a^{11}-\frac{1}{7}a^{10}-\frac{3}{7}a^{9}+\frac{1}{7}a^{8}-\frac{3}{7}a^{7}+\frac{3}{7}a^{6}+\frac{1}{7}a^{5}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}-\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{77}a^{13}-\frac{1}{77}a^{12}-\frac{3}{77}a^{11}-\frac{2}{7}a^{10}-\frac{1}{11}a^{9}+\frac{37}{77}a^{8}-\frac{26}{77}a^{7}-\frac{12}{77}a^{6}-\frac{9}{77}a^{5}-\frac{17}{77}a^{4}+\frac{3}{7}a^{3}+\frac{6}{77}a^{2}-\frac{9}{77}a+\frac{9}{77}$ 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:   \( -1 \)  (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:   $\frac{53}{77}a^{13}-\frac{306}{77}a^{12}+\frac{897}{77}a^{11}-\frac{153}{7}a^{10}+\frac{2236}{77}a^{9}-\frac{306}{11}a^{8}+\frac{1537}{77}a^{7}-\frac{779}{77}a^{6}+\frac{194}{77}a^{5}+\frac{254}{77}a^{4}-\frac{52}{7}a^{3}+\frac{571}{77}a^{2}-\frac{180}{77}a+\frac{37}{77}$, $\frac{3}{7}a^{12}-\frac{18}{7}a^{11}+\frac{53}{7}a^{10}-\frac{100}{7}a^{9}+\frac{136}{7}a^{8}-\frac{142}{7}a^{7}+\frac{121}{7}a^{6}-\frac{88}{7}a^{5}+8a^{4}-\frac{30}{7}a^{3}+\frac{4}{7}a^{2}+\frac{5}{7}a+\frac{4}{7}$, $\frac{46}{77}a^{13}-\frac{288}{77}a^{12}+\frac{929}{77}a^{11}-26a^{10}+\frac{3253}{77}a^{9}-\frac{4238}{77}a^{8}+\frac{4689}{77}a^{7}-\frac{4512}{77}a^{6}+\frac{3810}{77}a^{5}-\frac{2784}{77}a^{4}+\frac{148}{7}a^{3}-\frac{113}{11}a^{2}+\frac{389}{77}a-\frac{114}{77}$, $\frac{37}{77}a^{13}-\frac{32}{11}a^{12}+\frac{703}{77}a^{11}-\frac{134}{7}a^{10}+\frac{2304}{77}a^{9}-\frac{2822}{77}a^{8}+\frac{2833}{77}a^{7}-\frac{2391}{77}a^{6}+\frac{1713}{77}a^{5}-\frac{1014}{77}a^{4}+\frac{29}{7}a^{3}+\frac{24}{77}a^{2}-\frac{80}{77}a+\frac{58}{77}$, $\frac{2}{77}a^{13}-\frac{79}{77}a^{12}+\frac{456}{77}a^{11}-\frac{130}{7}a^{10}+\frac{427}{11}a^{9}-\frac{4623}{77}a^{8}+\frac{5569}{77}a^{7}-\frac{5568}{77}a^{6}+\frac{4756}{77}a^{5}-\frac{3576}{77}a^{4}+\frac{202}{7}a^{3}-\frac{989}{77}a^{2}+\frac{213}{77}a-\frac{59}{77}$, $\frac{27}{77}a^{13}-\frac{181}{77}a^{12}+\frac{612}{77}a^{11}-\frac{124}{7}a^{10}+\frac{325}{11}a^{9}-\frac{3005}{77}a^{8}+\frac{3302}{77}a^{7}-\frac{3096}{77}a^{6}+\frac{2529}{77}a^{5}-\frac{1768}{77}a^{4}+\frac{95}{7}a^{3}-\frac{454}{77}a^{2}+\frac{219}{77}a-\frac{65}{77}$ 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:  \( 2.56161916129 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 2.56161916129 \cdot 1}{2\cdot\sqrt{9095120158391}}\cr\approx \mathstrut & 0.164187248733 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 25*x^12 - 59*x^11 + 103*x^10 - 141*x^9 + 159*x^8 - 153*x^7 + 129*x^6 - 95*x^5 + 58*x^4 - 27*x^3 + 10*x^2 - 3*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 - 7*x^13 + 25*x^12 - 59*x^11 + 103*x^10 - 141*x^9 + 159*x^8 - 153*x^7 + 129*x^6 - 95*x^5 + 58*x^4 - 27*x^3 + 10*x^2 - 3*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 - 7*x^13 + 25*x^12 - 59*x^11 + 103*x^10 - 141*x^9 + 159*x^8 - 153*x^7 + 129*x^6 - 95*x^5 + 58*x^4 - 27*x^3 + 10*x^2 - 3*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 - 7*x^13 + 25*x^12 - 59*x^11 + 103*x^10 - 141*x^9 + 159*x^8 - 153*x^7 + 129*x^6 - 95*x^5 + 58*x^4 - 27*x^3 + 10*x^2 - 3*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

$D_7$ (as 14T2):

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 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-71}) \), 7.1.357911.1 x7

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 7 sibling: 7.1.357911.1
Minimal sibling: 7.1.357911.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }^{2}$ ${\href{/padicField/3.7.0.1}{7} }^{2}$ ${\href{/padicField/5.7.0.1}{7} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{7}$ ${\href{/padicField/11.2.0.1}{2} }^{7}$ ${\href{/padicField/13.2.0.1}{2} }^{7}$ ${\href{/padicField/17.2.0.1}{2} }^{7}$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{7}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{7}$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{7}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{7}$ ${\href{/padicField/53.2.0.1}{2} }^{7}$ ${\href{/padicField/59.2.0.1}{2} }^{7}$

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 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}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$