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Properties

Label 14.0.26456876306...1603.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,827^{7}$
Root discriminant $28.76$
Ramified prime $827$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![400, -1600, 3828, -3056, 557, -410, 1863, -1232, 473, -226, 113, -40, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 7*x^12 - 40*x^11 + 113*x^10 - 226*x^9 + 473*x^8 - 1232*x^7 + 1863*x^6 - 410*x^5 + 557*x^4 - 3056*x^3 + 3828*x^2 - 1600*x + 400)
 
gp: K = bnfinit(x^14 - 2*x^13 + 7*x^12 - 40*x^11 + 113*x^10 - 226*x^9 + 473*x^8 - 1232*x^7 + 1863*x^6 - 410*x^5 + 557*x^4 - 3056*x^3 + 3828*x^2 - 1600*x + 400, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 7 x^{12} - 40 x^{11} + 113 x^{10} - 226 x^{9} + 473 x^{8} - 1232 x^{7} + 1863 x^{6} - 410 x^{5} + 557 x^{4} - 3056 x^{3} + 3828 x^{2} - 1600 x + 400 \)

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:  \(-264568763060045171603=-\,827^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.76$
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:  $827$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{40} a^{9} + \frac{1}{20} a^{8} + \frac{1}{40} a^{7} - \frac{9}{40} a^{6} - \frac{1}{8} a^{5} - \frac{1}{40} a^{4} + \frac{3}{20} a^{3} + \frac{13}{40} a^{2} - \frac{7}{20} a - \frac{1}{2}$, $\frac{1}{40} a^{10} - \frac{3}{40} a^{8} - \frac{1}{40} a^{7} + \frac{3}{40} a^{6} - \frac{1}{40} a^{5} - \frac{1}{20} a^{4} - \frac{9}{40} a^{3} - \frac{1}{4} a^{2} - \frac{3}{10} a$, $\frac{1}{40} a^{11} - \frac{1}{8} a^{8} - \frac{1}{10} a^{7} + \frac{1}{20} a^{6} - \frac{7}{40} a^{5} - \frac{1}{20} a^{4} - \frac{1}{20} a^{3} + \frac{7}{40} a^{2} + \frac{9}{20} a - \frac{1}{2}$, $\frac{1}{280} a^{12} - \frac{3}{280} a^{11} - \frac{3}{280} a^{9} + \frac{5}{56} a^{8} - \frac{1}{70} a^{7} - \frac{11}{280} a^{6} + \frac{9}{280} a^{5} - \frac{19}{140} a^{4} - \frac{1}{8} a^{3} + \frac{13}{280} a^{2} - \frac{61}{140} a - \frac{5}{14}$, $\frac{1}{434837829484240} a^{13} - \frac{20696655139}{43483782948424} a^{12} + \frac{3880225274789}{434837829484240} a^{11} + \frac{1757578789157}{217418914742120} a^{10} + \frac{408754191029}{62119689926320} a^{9} + \frac{20259186072877}{217418914742120} a^{8} + \frac{33416373019}{12423937985264} a^{7} + \frac{173942771643}{10870945737106} a^{6} + \frac{57238724896621}{434837829484240} a^{5} + \frac{5519281403424}{27177364342765} a^{4} + \frac{91789323388671}{434837829484240} a^{3} - \frac{3415776651037}{54354728685530} a^{2} - \frac{13514623485733}{54354728685530} a + \frac{3599685333663}{10870945737106}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 114728.182626 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-827}) \), 7.1.565609283.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.565609283.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
827Data not computed