Properties

Label 14.2.19891027786401117.1
Degree $14$
Signature $[2, 6]$
Discriminant $1.989\times 10^{16}$
Root discriminant $14.59$
Ramified primes $3, 71$
Class number $1$
Class group trivial
Galois group $D_{14}$ (as 14T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 4*x^12 + 4*x^11 - 23*x^10 + 60*x^9 - 28*x^8 - 144*x^7 + 179*x^6 + 44*x^5 - 101*x^4 + 3*x^3 - 35*x^2 + 40*x - 11)
 
gp: K = bnfinit(x^14 - 4*x^13 + 4*x^12 + 4*x^11 - 23*x^10 + 60*x^9 - 28*x^8 - 144*x^7 + 179*x^6 + 44*x^5 - 101*x^4 + 3*x^3 - 35*x^2 + 40*x - 11, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 40, -35, 3, -101, 44, 179, -144, -28, 60, -23, 4, 4, -4, 1]);
 

\(x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 23 x^{10} + 60 x^{9} - 28 x^{8} - 144 x^{7} + 179 x^{6} + 44 x^{5} - 101 x^{4} + 3 x^{3} - 35 x^{2} + 40 x - 11\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(19891027786401117\)\(\medspace = 3^{7}\cdot 71^{7}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $14.59$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 71$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{371} a^{12} + \frac{87}{371} a^{11} + \frac{69}{371} a^{10} - \frac{137}{371} a^{9} - \frac{4}{371} a^{8} - \frac{109}{371} a^{7} - \frac{57}{371} a^{6} - \frac{166}{371} a^{5} + \frac{51}{371} a^{4} - \frac{29}{371} a^{3} + \frac{85}{371} a^{2} - \frac{139}{371} a - \frac{61}{371}$, $\frac{1}{13851723893} a^{13} - \frac{11895074}{13851723893} a^{12} + \frac{46708456}{446829803} a^{11} + \frac{4285957859}{13851723893} a^{10} + \frac{3471345978}{13851723893} a^{9} - \frac{1060103777}{13851723893} a^{8} + \frac{6251491559}{13851723893} a^{7} - \frac{435169327}{1978817699} a^{6} + \frac{4740766841}{13851723893} a^{5} + \frac{2886994029}{13851723893} a^{4} + \frac{3662836869}{13851723893} a^{3} + \frac{2833883202}{13851723893} a^{2} - \frac{2756620744}{13851723893} a - \frac{66905649}{261353281}$  Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 278.626564509 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{6}\cdot 278.626564509 \cdot 1}{2\sqrt{19891027786401117}}\approx 0.243110162641$

Galois group

$D_{14}$ (as 14T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 28
The 10 conjugacy class representatives for $D_{14}$
Character table for $D_{14}$

Intermediate fields

\(\Q(\sqrt{213}) \), 7.1.357911.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: Deg 28
Degree 14 sibling: 14.0.280155320935227.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
71Data not computed