Properties

Label 14.2.729722900774101.1
Degree $14$
Signature $[2, 6]$
Discriminant $7.297\times 10^{14}$
Root discriminant \(11.53\)
Ramified primes $13,109,181$
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 - 3*x^12 - x^11 + 10*x^10 - x^9 - 10*x^8 + 2*x^7 + 10*x^6 + 5*x^5 - 4*x^4 - 2*x^3 + 3*x^2 + 2*x - 1)
 
gp: K = bnfinit(y^14 - 3*y^12 - y^11 + 10*y^10 - y^9 - 10*y^8 + 2*y^7 + 10*y^6 + 5*y^5 - 4*y^4 - 2*y^3 + 3*y^2 + 2*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^12 - x^11 + 10*x^10 - x^9 - 10*x^8 + 2*x^7 + 10*x^6 + 5*x^5 - 4*x^4 - 2*x^3 + 3*x^2 + 2*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^12 - x^11 + 10*x^10 - x^9 - 10*x^8 + 2*x^7 + 10*x^6 + 5*x^5 - 4*x^4 - 2*x^3 + 3*x^2 + 2*x - 1)
 

\( x^{14} - 3 x^{12} - x^{11} + 10 x^{10} - x^{9} - 10 x^{8} + 2 x^{7} + 10 x^{6} + 5 x^{5} - 4 x^{4} - 2 x^{3} + 3 x^{2} + 2 x - 1 \) 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:   \(729722900774101\) \(\medspace = 13^{4}\cdot 109^{4}\cdot 181\) 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:  \(11.53\)
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:  $13^{1/2}109^{1/2}181^{1/2}\approx 506.43558326800064$
Ramified primes:   \(13\), \(109\), \(181\) 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:  \(\Q(\sqrt{181}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{426319}a^{13}+\frac{150375}{426319}a^{12}-\frac{171776}{426319}a^{11}-\frac{147791}{426319}a^{10}-\frac{62145}{426319}a^{9}-\frac{141896}{426319}a^{8}+\frac{81259}{426319}a^{7}+\frac{166949}{426319}a^{6}-\frac{117387}{426319}a^{5}+\frac{94394}{426319}a^{4}+\frac{206641}{426319}a^{3}+\frac{101101}{426319}a^{2}+\frac{101019}{426319}a+\frac{133519}{426319}$ 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:   $\frac{46977}{426319}a^{13}+\frac{60545}{426319}a^{12}-\frac{155120}{426319}a^{11}-\frac{172892}{426319}a^{10}+\frac{473166}{426319}a^{9}+\frac{501811}{426319}a^{8}-\frac{808921}{426319}a^{7}-\frac{227470}{426319}a^{6}+\frac{799804}{426319}a^{5}+\frac{1055657}{426319}a^{4}-\frac{335692}{426319}a^{3}-\frac{198302}{426319}a^{2}+\frac{639093}{426319}a+\frac{316935}{426319}$, $\frac{12531}{426319}a^{13}+\frac{19145}{426319}a^{12}-\frac{40425}{426319}a^{11}-\frac{39285}{426319}a^{10}+\frac{145818}{426319}a^{9}+\frac{77773}{426319}a^{8}-\frac{219562}{426319}a^{7}+\frac{90586}{426319}a^{6}+\frac{250372}{426319}a^{5}-\frac{184011}{426319}a^{4}+\frac{383084}{426319}a^{3}-\frac{123437}{426319}a^{2}+\frac{127978}{426319}a+\frac{250833}{426319}$, $a$, $\frac{44371}{426319}a^{13}-\frac{29544}{426319}a^{12}-\frac{141814}{426319}a^{11}+\frac{4397}{426319}a^{10}+\frac{421816}{426319}a^{9}-\frac{188424}{426319}a^{8}-\frac{263013}{426319}a^{7}-\frac{24865}{426319}a^{6}-\frac{239354}{426319}a^{5}+\frac{624637}{426319}a^{4}+\frac{25078}{426319}a^{3}-\frac{628685}{426319}a^{2}-\frac{430236}{426319}a-\frac{183594}{426319}$, $\frac{188630}{426319}a^{13}+\frac{101585}{426319}a^{12}-\frac{583923}{426319}a^{11}-\frac{390601}{426319}a^{10}+\frac{1787469}{426319}a^{9}+\frac{595935}{426319}a^{8}-\frac{2119751}{426319}a^{7}+\frac{257978}{426319}a^{6}+\frac{1578007}{426319}a^{5}+\frac{1606142}{426319}a^{4}+\frac{345660}{426319}a^{3}-\frac{698835}{426319}a^{2}+\frac{459946}{426319}a+\frac{467726}{426319}$, $\frac{190288}{426319}a^{13}+\frac{26720}{426319}a^{12}-\frac{607439}{426319}a^{11}-\frac{294654}{426319}a^{10}+\frac{1920257}{426319}a^{9}+\frac{234136}{426319}a^{8}-\frac{2109133}{426319}a^{7}-\frac{47930}{426319}a^{6}+\frac{1778144}{426319}a^{5}+\frac{1652321}{426319}a^{4}-\frac{1082995}{426319}a^{3}-\frac{616744}{426319}a^{2}+\frac{406081}{426319}a+\frac{156348}{426319}$, $\frac{216005}{426319}a^{13}+\frac{80946}{426319}a^{12}-\frac{653353}{426319}a^{11}-\frac{401916}{426319}a^{10}+\frac{2007223}{426319}a^{9}+\frac{385344}{426319}a^{8}-\frac{2187168}{426319}a^{7}+\frac{347173}{426319}a^{6}+\frac{1701504}{426319}a^{5}+\frac{1296114}{426319}a^{4}-\frac{110095}{426319}a^{3}-\frac{295589}{426319}a^{2}+\frac{323718}{426319}a+\frac{291245}{426319}$ 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:  \( 44.6471365658 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 44.6471365658 \cdot 1}{2\cdot\sqrt{729722900774101}}\cr\approx \mathstrut & 0.203387298874 \end{aligned}\]

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

$C_2^7.\GL(3,2)$ (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

7.3.2007889.1

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 ${\href{/padicField/2.14.0.1}{14} }$ ${\href{/padicField/3.7.0.1}{7} }^{2}$ ${\href{/padicField/5.7.0.1}{7} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ R ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(13\) Copy content Toggle raw display 13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 156 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 156 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
\(109\) Copy content Toggle raw display $\Q_{109}$$x + 103$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$[\ ]$
109.4.2.1$x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.0.1$x^{4} + 11 x^{2} + 98 x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
109.4.2.1$x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(181\) Copy content Toggle raw display 181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.4.0.1$x^{4} + 6 x^{2} + 105 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
181.4.0.1$x^{4} + 6 x^{2} + 105 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
181.4.0.1$x^{4} + 6 x^{2} + 105 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$