Properties

Label 14.4.498897726810659.1
Degree $14$
Signature $[4, 5]$
Discriminant $-4.989\times 10^{14}$
Root discriminant \(11.22\)
Ramified primes $11,71,8623$
Class number $1$
Class group trivial
Galois group $C_2^7.S_7$ (as 14T57)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{14} - 2x^{13} + 3x^{11} - 7x^{10} + x^{9} + 14x^{8} - 2x^{7} - 13x^{6} + 2x^{5} + 4x^{4} - 4x^{3} + 3x + 1 \) 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:  $[4, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-498897726810659\) \(\medspace = -\,11^{3}\cdot 71^{2}\cdot 8623^{2}\) 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:  \(11.22\)
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:  $11^{1/2}71^{1/2}8623^{1/2}\approx 2595.103658815963$
Ramified primes:   \(11\), \(71\), \(8623\) 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:  \(\Q(\sqrt{-11}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{31}a^{13}+\frac{3}{31}a^{12}+\frac{15}{31}a^{11}-\frac{15}{31}a^{10}+\frac{11}{31}a^{9}-\frac{6}{31}a^{8}+\frac{15}{31}a^{7}+\frac{11}{31}a^{6}+\frac{11}{31}a^{5}-\frac{5}{31}a^{4}+\frac{10}{31}a^{3}+\frac{15}{31}a^{2}+\frac{13}{31}a+\frac{6}{31}$ 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:  $8$
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:   $\frac{54}{31}a^{13}-\frac{117}{31}a^{12}+\frac{35}{31}a^{11}+\frac{151}{31}a^{10}-\frac{429}{31}a^{9}+\frac{172}{31}a^{8}+\frac{655}{31}a^{7}-\frac{336}{31}a^{6}-\frac{491}{31}a^{5}+\frac{288}{31}a^{4}+\frac{75}{31}a^{3}-\frac{244}{31}a^{2}+\frac{113}{31}a+\frac{107}{31}$, $\frac{58}{31}a^{13}-\frac{136}{31}a^{12}+\frac{64}{31}a^{11}+\frac{122}{31}a^{10}-\frac{447}{31}a^{9}+\frac{241}{31}a^{8}+\frac{622}{31}a^{7}-\frac{323}{31}a^{6}-\frac{478}{31}a^{5}+\frac{330}{31}a^{4}+\frac{22}{31}a^{3}-\frac{246}{31}a^{2}+\frac{103}{31}a+\frac{100}{31}$, $\frac{25}{31}a^{13}-\frac{80}{31}a^{12}+\frac{34}{31}a^{11}+\frac{90}{31}a^{10}-\frac{252}{31}a^{9}+\frac{191}{31}a^{8}+\frac{437}{31}a^{7}-\frac{345}{31}a^{6}-\frac{407}{31}a^{5}+\frac{309}{31}a^{4}+\frac{64}{31}a^{3}-\frac{214}{31}a^{2}+\frac{108}{31}a+\frac{88}{31}$, $\frac{46}{31}a^{13}-\frac{110}{31}a^{12}+\frac{101}{31}a^{11}+\frac{23}{31}a^{10}-\frac{331}{31}a^{9}+\frac{282}{31}a^{8}+\frac{194}{31}a^{7}-\frac{238}{31}a^{6}-\frac{52}{31}a^{5}+\frac{235}{31}a^{4}-\frac{129}{31}a^{3}-\frac{54}{31}a^{2}+\frac{71}{31}a-\frac{34}{31}$, $\frac{88}{31}a^{13}-\frac{232}{31}a^{12}+\frac{142}{31}a^{11}+\frac{168}{31}a^{10}-\frac{706}{31}a^{9}+\frac{526}{31}a^{8}+\frac{917}{31}a^{7}-\frac{675}{31}a^{6}-\frac{737}{31}a^{5}+\frac{552}{31}a^{4}-\frac{19}{31}a^{3}-\frac{354}{31}a^{2}+\frac{183}{31}a+\frac{156}{31}$, $\frac{30}{31}a^{13}-\frac{34}{31}a^{12}+\frac{16}{31}a^{11}+\frac{15}{31}a^{10}-\frac{135}{31}a^{9}-\frac{56}{31}a^{8}+\frac{109}{31}a^{7}+\frac{144}{31}a^{6}-\frac{11}{31}a^{5}+\frac{36}{31}a^{4}-\frac{41}{31}a^{3}-\frac{46}{31}a^{2}-\frac{75}{31}a-\frac{6}{31}$, $\frac{53}{31}a^{13}-\frac{151}{31}a^{12}+\frac{82}{31}a^{11}+\frac{135}{31}a^{10}-\frac{471}{31}a^{9}+\frac{364}{31}a^{8}+\frac{671}{31}a^{7}-\frac{533}{31}a^{6}-\frac{564}{31}a^{5}+\frac{417}{31}a^{4}+\frac{65}{31}a^{3}-\frac{259}{31}a^{2}+\frac{100}{31}a+\frac{101}{31}$, $\frac{1}{31}a^{13}+\frac{3}{31}a^{12}+\frac{15}{31}a^{11}-\frac{46}{31}a^{10}+\frac{42}{31}a^{9}-\frac{6}{31}a^{8}-\frac{140}{31}a^{7}+\frac{135}{31}a^{6}+\frac{135}{31}a^{5}-\frac{129}{31}a^{4}-\frac{83}{31}a^{3}+\frac{108}{31}a^{2}-\frac{18}{31}a-\frac{56}{31}$ 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:  \( 47.3515916708 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{5}\cdot 47.3515916708 \cdot 1}{2\cdot\sqrt{498897726810659}}\cr\approx \mathstrut & 0.166080310507 \end{aligned}\]

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

$C_2^7.S_7$ (as 14T57):

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 non-solvable group of order 645120
The 110 conjugacy class representatives for $C_2^7.S_7$ are not computed
Character table for $C_2^7.S_7$ is not computed

Intermediate fields

7.3.612233.1

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 42 siblings: 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 ${\href{/padicField/2.14.0.1}{14} }$ ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ R ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# 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
\(11\) Copy content Toggle raw display 11.6.3.1$x^{6} + 242 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.8.0.1$x^{8} + 7 x^{4} + 7 x^{3} + x^{2} + 7 x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
\(71\) Copy content Toggle raw display 71.2.1.1$x^{2} + 497$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} + 497$$2$$1$$1$$C_2$$[\ ]_{2}$
71.5.0.1$x^{5} + 18 x + 64$$1$$5$$0$$C_5$$[\ ]^{5}$
71.5.0.1$x^{5} + 18 x + 64$$1$$5$$0$$C_5$$[\ ]^{5}$
\(8623\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$