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Properties

Label 14.2.959588563913537.1
Degree $14$
Signature $[2, 6]$
Discriminant $9.596\times 10^{14}$
Root discriminant \(11.75\)
Ramified primes $17,2741$
Class number $1$
Class group trivial
Galois group $C_2^7.\GL(3,2)$ (as 14T51)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 15*x^12 - 18*x^11 - x^10 + 52*x^9 - 125*x^8 + 189*x^7 - 217*x^6 + 198*x^5 - 146*x^4 + 83*x^3 - 33*x^2 + 8*x - 1)
 
gp: K = bnfinit(y^14 - 6*y^13 + 15*y^12 - 18*y^11 - y^10 + 52*y^9 - 125*y^8 + 189*y^7 - 217*y^6 + 198*y^5 - 146*y^4 + 83*y^3 - 33*y^2 + 8*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 6*x^13 + 15*x^12 - 18*x^11 - x^10 + 52*x^9 - 125*x^8 + 189*x^7 - 217*x^6 + 198*x^5 - 146*x^4 + 83*x^3 - 33*x^2 + 8*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 6*x^13 + 15*x^12 - 18*x^11 - x^10 + 52*x^9 - 125*x^8 + 189*x^7 - 217*x^6 + 198*x^5 - 146*x^4 + 83*x^3 - 33*x^2 + 8*x - 1)
 

\( x^{14} - 6 x^{13} + 15 x^{12} - 18 x^{11} - x^{10} + 52 x^{9} - 125 x^{8} + 189 x^{7} - 217 x^{6} + \cdots - 1 \) 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:   \(959588563913537\) \(\medspace = 17\cdot 2741^{4}\) 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:  \(11.75\)
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:  $17^{1/2}2741^{1/2}\approx 215.8633827215723$
Ramified primes:   \(17\), \(2741\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{8669}a^{13}-\frac{226}{8669}a^{12}-\frac{2279}{8669}a^{11}-\frac{1440}{8669}a^{10}-\frac{3954}{8669}a^{9}+\frac{3032}{8669}a^{8}+\frac{348}{8669}a^{7}+\frac{1650}{8669}a^{6}+\frac{881}{8669}a^{5}-\frac{2904}{8669}a^{4}-\frac{2772}{8669}a^{3}+\frac{3093}{8669}a^{2}-\frac{4311}{8669}a+\frac{3507}{8669}$ 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:   $\frac{12708}{8669}a^{13}-\frac{71921}{8669}a^{12}+\frac{166308}{8669}a^{11}-\frac{172641}{8669}a^{10}-\frac{71260}{8669}a^{9}+\frac{638457}{8669}a^{8}-\frac{1377177}{8669}a^{7}+\frac{1939745}{8669}a^{6}-\frac{2093829}{8669}a^{5}+\frac{1785715}{8669}a^{4}-\frac{1226758}{8669}a^{3}+\frac{616097}{8669}a^{2}-\frac{195495}{8669}a+\frac{25634}{8669}$, $\frac{4937}{8669}a^{13}-\frac{14799}{8669}a^{12}+\frac{939}{8669}a^{11}+\frac{51314}{8669}a^{10}-\frac{102338}{8669}a^{9}+\frac{101649}{8669}a^{8}+\frac{10283}{8669}a^{7}-\frac{202197}{8669}a^{6}+\frac{344419}{8669}a^{5}-\frac{431972}{8669}a^{4}+\frac{367085}{8669}a^{3}-\frac{273376}{8669}a^{2}+\frac{111685}{8669}a-\frac{23941}{8669}$, $a$, $\frac{6525}{8669}a^{13}-\frac{35596}{8669}a^{12}+\frac{74881}{8669}a^{11}-\frac{59487}{8669}a^{10}-\frac{70258}{8669}a^{9}+\frac{321895}{8669}a^{8}-\frac{607408}{8669}a^{7}+\frac{762224}{8669}a^{6}-\frac{735887}{8669}a^{5}+\frac{556650}{8669}a^{4}-\frac{315850}{8669}a^{3}+\frac{113090}{8669}a^{2}+\frac{1630}{8669}a-\frac{2985}{8669}$, $\frac{6525}{8669}a^{13}-\frac{35596}{8669}a^{12}+\frac{74881}{8669}a^{11}-\frac{59487}{8669}a^{10}-\frac{70258}{8669}a^{9}+\frac{321895}{8669}a^{8}-\frac{607408}{8669}a^{7}+\frac{762224}{8669}a^{6}-\frac{735887}{8669}a^{5}+\frac{556650}{8669}a^{4}-\frac{315850}{8669}a^{3}+\frac{113090}{8669}a^{2}+\frac{1630}{8669}a-\frac{11654}{8669}$, $\frac{6915}{8669}a^{13}-\frac{45715}{8669}a^{12}+\frac{122323}{8669}a^{11}-\frac{152961}{8669}a^{10}+\frac{116}{8669}a^{9}+\frac{412081}{8669}a^{8}-\frac{1009166}{8669}a^{7}+\frac{1535759}{8669}a^{6}-\frac{1753330}{8669}a^{5}+\frac{1600009}{8669}a^{4}-\frac{1162867}{8669}a^{3}+\frac{651847}{8669}a^{2}-\frac{231937}{8669}a+\frac{38388}{8669}$, $\frac{6154}{8669}a^{13}-\frac{29771}{8669}a^{12}+\frac{53490}{8669}a^{11}-\frac{28049}{8669}a^{10}-\frac{77054}{8669}a^{9}+\frac{254641}{8669}a^{8}-\frac{424432}{8669}a^{7}+\frac{488165}{8669}a^{6}-\frac{455908}{8669}a^{5}+\frac{325015}{8669}a^{4}-\frac{180345}{8669}a^{3}+\frac{57881}{8669}a^{2}+\frac{5915}{8669}a-\frac{3732}{8669}$ 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:  \( 60.8997778777 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 60.8997778777 \cdot 1}{2\cdot\sqrt{959588563913537}}\cr\approx \mathstrut & 0.241925909634 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 15*x^12 - 18*x^11 - x^10 + 52*x^9 - 125*x^8 + 189*x^7 - 217*x^6 + 198*x^5 - 146*x^4 + 83*x^3 - 33*x^2 + 8*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 - 6*x^13 + 15*x^12 - 18*x^11 - x^10 + 52*x^9 - 125*x^8 + 189*x^7 - 217*x^6 + 198*x^5 - 146*x^4 + 83*x^3 - 33*x^2 + 8*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 - 6*x^13 + 15*x^12 - 18*x^11 - x^10 + 52*x^9 - 125*x^8 + 189*x^7 - 217*x^6 + 198*x^5 - 146*x^4 + 83*x^3 - 33*x^2 + 8*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 - 6*x^13 + 15*x^12 - 18*x^11 - x^10 + 52*x^9 - 125*x^8 + 189*x^7 - 217*x^6 + 198*x^5 - 146*x^4 + 83*x^3 - 33*x^2 + 8*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

$C_2^7.\GL(3,2)$ (as 14T51):

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 21504
The 48 conjugacy class representatives for $C_2^7.\GL(3,2)$
Character table for $C_2^7.\GL(3,2)$ is not computed

Intermediate fields

7.3.7513081.2

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 ${\href{/padicField/2.7.0.1}{7} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.14.0.1}{14} }$ ${\href{/padicField/13.7.0.1}{7} }^{2}$ R ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.7.0.1}{7} }^{2}$

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
\(17\) Copy content Toggle raw display 17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
\(2741\) Copy content Toggle raw display $\Q_{2741}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2741}$$x$$1$$1$$0$Trivial$[\ ]$
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 $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$