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Properties

Label 14.10.340...409.1
Degree $14$
Signature $[10, 2]$
Discriminant $3.404\times 10^{17}$
Root discriminant \(17.88\)
Ramified primes $17,23,64879$
Class number $1$
Class group trivial
Galois group $C_2^6.S_7$ (as 14T55)

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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 + 14*x^12 - 16*x^10 - 14*x^9 + 9*x^8 + 71*x^7 - 64*x^6 - 9*x^5 + 14*x^4 - 5*x^3 + 10*x^2 - 6*x + 1)
 
gp: K = bnfinit(y^14 - 7*y^13 + 14*y^12 - 16*y^10 - 14*y^9 + 9*y^8 + 71*y^7 - 64*y^6 - 9*y^5 + 14*y^4 - 5*y^3 + 10*y^2 - 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 7*x^13 + 14*x^12 - 16*x^10 - 14*x^9 + 9*x^8 + 71*x^7 - 64*x^6 - 9*x^5 + 14*x^4 - 5*x^3 + 10*x^2 - 6*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 7*x^13 + 14*x^12 - 16*x^10 - 14*x^9 + 9*x^8 + 71*x^7 - 64*x^6 - 9*x^5 + 14*x^4 - 5*x^3 + 10*x^2 - 6*x + 1)
 

\( x^{14} - 7 x^{13} + 14 x^{12} - 16 x^{10} - 14 x^{9} + 9 x^{8} + 71 x^{7} - 64 x^{6} - 9 x^{5} + \cdots + 1 \) 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:  $[10, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(340421892311181409\) \(\medspace = 17^{2}\cdot 23^{4}\cdot 64879^{2}\) 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:  \(17.88\)
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:  $17^{1/2}23^{3/4}64879^{1/2}\approx 11029.921240869915$
Ramified primes:   \(17\), \(23\), \(64879\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{829}a^{13}+\frac{133}{829}a^{12}+\frac{396}{829}a^{11}-\frac{103}{829}a^{10}-\frac{343}{829}a^{9}+\frac{48}{829}a^{8}+\frac{97}{829}a^{7}+\frac{387}{829}a^{6}+\frac{231}{829}a^{5}+\frac{14}{829}a^{3}+\frac{297}{829}a^{2}+\frac{140}{829}a-\frac{302}{829}$ 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:  $11$
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:   $\frac{18244}{829}a^{13}-\frac{120236}{829}a^{12}+\frac{207139}{829}a^{11}+\frac{79795}{829}a^{10}-\frac{254903}{829}a^{9}-\frac{351208}{829}a^{8}+\frac{20478}{829}a^{7}+\frac{1287272}{829}a^{6}-\frac{662643}{829}a^{5}-486a^{4}+\frac{95419}{829}a^{3}-\frac{54590}{829}a^{2}+\frac{160008}{829}a-\frac{48236}{829}$, $a$, $\frac{19755}{829}a^{13}-\frac{130668}{829}a^{12}+\frac{226853}{829}a^{11}+\frac{84159}{829}a^{10}-\frac{281579}{829}a^{9}-\frac{378160}{829}a^{8}+\frac{33576}{829}a^{7}+\frac{1403644}{829}a^{6}-\frac{742195}{829}a^{5}-537a^{4}+\frac{115744}{829}a^{3}-\frac{54312}{829}a^{2}+\frac{175904}{829}a-\frac{54411}{829}$, $a-1$, $\frac{24352}{829}a^{13}-\frac{160913}{829}a^{12}+\frac{278179}{829}a^{11}+\frac{107239}{829}a^{10}-\frac{346254}{829}a^{9}-\frac{472524}{829}a^{8}+\frac{33483}{829}a^{7}+\frac{1734420}{829}a^{6}-\frac{887312}{829}a^{5}-667a^{4}+\frac{130362}{829}a^{3}-\frac{75920}{829}a^{2}+\frac{213485}{829}a-\frac{64078}{829}$, $\frac{7617}{829}a^{13}-\frac{49717}{829}a^{12}+\frac{84159}{829}a^{11}+\frac{34501}{829}a^{10}-\frac{101590}{829}a^{9}-\frac{144219}{829}a^{8}+\frac{1039}{829}a^{7}+\frac{522125}{829}a^{6}-\frac{267378}{829}a^{5}-194a^{4}+\frac{44463}{829}a^{3}-\frac{21646}{829}a^{2}+\frac{64948}{829}a-\frac{20584}{829}$, $\frac{8836}{829}a^{13}-\frac{57535}{829}a^{12}+\frac{96011}{829}a^{11}+\frac{44900}{829}a^{10}-\frac{118471}{829}a^{9}-\frac{176897}{829}a^{8}-\frac{5897}{829}a^{7}+\frac{617512}{829}a^{6}-\frac{275939}{829}a^{5}-245a^{4}+\frac{32514}{829}a^{3}-\frac{30995}{829}a^{2}+\frac{73124}{829}a-\frac{19817}{829}$, $\frac{7151}{829}a^{13}-\frac{47862}{829}a^{12}+\frac{85319}{829}a^{11}+\frac{26956}{829}a^{10}-\frac{107552}{829}a^{9}-\frac{134256}{829}a^{8}+\frac{26302}{829}a^{7}+\frac{519189}{829}a^{6}-\frac{299585}{829}a^{5}-203a^{4}+\frac{51203}{829}a^{3}-\frac{11657}{829}a^{2}+\frac{66028}{829}a-\frac{21611}{829}$, $\frac{22271}{829}a^{13}-\frac{145878}{829}a^{12}+\frac{247456}{829}a^{11}+\frac{104384}{829}a^{10}-\frac{303132}{829}a^{9}-\frac{434798}{829}a^{8}+\frac{2400}{829}a^{7}+\frac{1554968}{829}a^{6}-\frac{757050}{829}a^{5}-588a^{4}+\frac{108689}{829}a^{3}-\frac{72227}{829}a^{2}+\frac{188254}{829}a-\frac{54879}{829}$, $\frac{9653}{829}a^{13}-\frac{63276}{829}a^{12}+\frac{107839}{829}a^{11}+\frac{43649}{829}a^{10}-\frac{131764}{829}a^{9}-\frac{185763}{829}a^{8}+\frac{5374}{829}a^{7}+\frac{673385}{829}a^{6}-\frac{340057}{829}a^{5}-256a^{4}+\frac{48926}{829}a^{3}-\frac{27098}{829}a^{2}+\frac{83879}{829}a-\frac{25312}{829}$, $\frac{22157}{829}a^{13}-\frac{146118}{829}a^{12}+\frac{251223}{829}a^{11}+\frac{100375}{829}a^{10}-\frac{312112}{829}a^{9}-\frac{434467}{829}a^{8}+\frac{22015}{829}a^{7}+\frac{1576341}{829}a^{6}-\frac{783384}{829}a^{5}-608a^{4}+\frac{106264}{829}a^{3}-\frac{70438}{829}a^{2}+\frac{192190}{829}a-\frac{56927}{829}$ 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:  \( 5068.09703458 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{2}\cdot 5068.09703458 \cdot 1}{2\cdot\sqrt{340421892311181409}}\cr\approx \mathstrut & 0.175576294179 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 14*x^12 - 16*x^10 - 14*x^9 + 9*x^8 + 71*x^7 - 64*x^6 - 9*x^5 + 14*x^4 - 5*x^3 + 10*x^2 - 6*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 + 14*x^12 - 16*x^10 - 14*x^9 + 9*x^8 + 71*x^7 - 64*x^6 - 9*x^5 + 14*x^4 - 5*x^3 + 10*x^2 - 6*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 + 14*x^12 - 16*x^10 - 14*x^9 + 9*x^8 + 71*x^7 - 64*x^6 - 9*x^5 + 14*x^4 - 5*x^3 + 10*x^2 - 6*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 + 14*x^12 - 16*x^10 - 14*x^9 + 9*x^8 + 71*x^7 - 64*x^6 - 9*x^5 + 14*x^4 - 5*x^3 + 10*x^2 - 6*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

$C_2^6.S_7$ (as 14T55):

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 322560
The 55 conjugacy class representatives for $C_2^6.S_7$
Character table for $C_2^6.S_7$

Intermediate fields

7.7.25367689.1

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 sibling: data not computed
Degree 42 siblings: data not computed
Minimal sibling: 14.10.6760305669422759337233840504353774605150973117780729.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }^{2}$ ${\href{/padicField/3.7.0.1}{7} }^{2}$ ${\href{/padicField/5.7.0.1}{7} }^{2}$ ${\href{/padicField/7.7.0.1}{7} }^{2}$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ R ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.0.1$x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
\(23\) Copy content Toggle raw display 23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.3.2$x^{4} + 115$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.6.0.1$x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
\(64879\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$