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Properties

Label 14.0.51902598731283.1
Degree $14$
Signature $[0, 7]$
Discriminant $-5.190\times 10^{13}$
Root discriminant $9.54$
Ramified primes $3, 17117$
Class number $1$
Class group trivial
Galois group $S_7\times C_2$ (as 14T49)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 3*x^12 - 6*x^11 + 6*x^10 - 4*x^9 + 3*x^8 + x^7 - x^5 + 3*x^4 + 3*x^2 + x + 1)
 
gp: K = bnfinit(x^14 - 2*x^13 + 3*x^12 - 6*x^11 + 6*x^10 - 4*x^9 + 3*x^8 + x^7 - x^5 + 3*x^4 + 3*x^2 + x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 3, 0, 3, -1, 0, 1, 3, -4, 6, -6, 3, -2, 1]);
 

\( x^{14} - 2 x^{13} + 3 x^{12} - 6 x^{11} + 6 x^{10} - 4 x^{9} + 3 x^{8} + x^{7} - x^{5} + 3 x^{4} + 3 x^{2} + x + 1 \)

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

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-51902598731283\)\(\medspace = -\,3^{11}\cdot 17117^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.54$
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:  $3, 17117$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{9} a^{11} - \frac{2}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{10} + \frac{7}{27} a^{9} - \frac{2}{9} a^{8} + \frac{1}{27} a^{7} - \frac{10}{27} a^{6} - \frac{8}{27} a^{5} + \frac{1}{3} a^{4} + \frac{10}{27} a^{3} - \frac{1}{27} a^{2} - \frac{4}{9} a - \frac{8}{27}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{11} + \frac{7}{27} a^{10} - \frac{2}{9} a^{9} + \frac{1}{27} a^{8} - \frac{10}{27} a^{7} - \frac{8}{27} a^{6} + \frac{1}{3} a^{5} + \frac{10}{27} a^{4} - \frac{1}{27} a^{3} - \frac{4}{9} a^{2} - \frac{8}{27} a$

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:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{10}{27} a^{13} - \frac{13}{27} a^{12} + \frac{17}{27} a^{11} - \frac{52}{27} a^{10} + \frac{38}{27} a^{9} - \frac{20}{27} a^{8} + \frac{49}{27} a^{7} - \frac{4}{27} a^{6} + \frac{5}{27} a^{5} - \frac{17}{27} a^{4} + \frac{22}{27} a^{3} + \frac{1}{27} a^{2} + \frac{49}{27} a + \frac{23}{27} \) (order $6$)
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:  \( 27.6082700412 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{7}\cdot 27.6082700412 \cdot 1}{6\sqrt{51902598731283}}\approx 0.246917840876$

Galois group

$S_7\times C_2$ (as 14T49):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 10080
The 30 conjugacy class representatives for $S_7\times C_2$
Character table for $S_7\times C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 7.1.462159.1

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

Sibling fields

Degree 14 sibling: data not computed
Degree 28 sibling: data not computed
Degree 42 siblings: data not computed

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 ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17117Data not computed