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Properties

Label 14.0.15192196831...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 5^{7}\cdot 7^{15}$
Root discriminant $32.58$
Ramified primes $2, 5, 7$
Class number $2$
Class group $[2]$
Galois group $F_7 \times C_2$ (as 14T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8635, -15855, 22526, -24689, 20615, -13006, 6965, -3805, 1869, -616, 91, -7, 14, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 91*x^10 - 616*x^9 + 1869*x^8 - 3805*x^7 + 6965*x^6 - 13006*x^5 + 20615*x^4 - 24689*x^3 + 22526*x^2 - 15855*x + 8635)
 
gp: K = bnfinit(x^14 - 7*x^13 + 14*x^12 - 7*x^11 + 91*x^10 - 616*x^9 + 1869*x^8 - 3805*x^7 + 6965*x^6 - 13006*x^5 + 20615*x^4 - 24689*x^3 + 22526*x^2 - 15855*x + 8635, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 14 x^{12} - 7 x^{11} + 91 x^{10} - 616 x^{9} + 1869 x^{8} - 3805 x^{7} + 6965 x^{6} - 13006 x^{5} + 20615 x^{4} - 24689 x^{3} + 22526 x^{2} - 15855 x + 8635 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1519219683181760000000=-\,2^{12}\cdot 5^{7}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.58$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} - \frac{1}{13} a^{9} + \frac{2}{13} a^{7} - \frac{5}{13} a^{6} - \frac{3}{13} a^{5} - \frac{5}{13} a^{4} + \frac{2}{13} a^{3} + \frac{2}{13} a^{2} + \frac{2}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{12} - \frac{1}{13} a^{10} + \frac{2}{13} a^{8} - \frac{5}{13} a^{7} - \frac{3}{13} a^{6} - \frac{5}{13} a^{5} + \frac{2}{13} a^{4} + \frac{2}{13} a^{3} + \frac{2}{13} a^{2} - \frac{3}{13} a$, $\frac{1}{688470099609061871049} a^{13} - \frac{21112672754012246660}{688470099609061871049} a^{12} - \frac{22831577843023664681}{688470099609061871049} a^{11} + \frac{85854245890798038322}{688470099609061871049} a^{10} + \frac{56942517430159755708}{229490033203020623683} a^{9} - \frac{293570460697681588750}{688470099609061871049} a^{8} - \frac{49303451362355378897}{688470099609061871049} a^{7} + \frac{332206980343729876756}{688470099609061871049} a^{6} + \frac{99117383095477125085}{688470099609061871049} a^{5} + \frac{98429847036096947407}{688470099609061871049} a^{4} + \frac{300039762854757732724}{688470099609061871049} a^{3} + \frac{67570752602887919204}{229490033203020623683} a^{2} + \frac{156780569667322162052}{688470099609061871049} a + \frac{340949388951406806076}{688470099609061871049}$

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

Class group and class number

$C_{2}$, which has order $2$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 98751.2336328229 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_7$ (as 14T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 84
The 14 conjugacy class representatives for $F_7 \times C_2$
Character table for $F_7 \times C_2$

Intermediate fields

\(\Q(\sqrt{-35}) \), 7.1.52706752.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.14.12.1$x^{14} - 2 x^{7} + 4$$7$$2$$12$$(C_7:C_3) \times C_2$$[\ ]_{7}^{6}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.14.15.1$x^{14} + 21 x^{13} + 14 x^{12} + 7 x^{11} - 14 x^{10} - 7 x^{9} - 7 x^{8} - 7 x^{7} - 7 x^{6} - 21 x^{5} - 7 x^{4} - 14 x^{3} + 7 x^{2} + 21$$14$$1$$15$$F_7 \times C_2$$[7/6]_{6}^{2}$