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Properties

Label 14.0.89053868635...6875.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 7^{19}$
Root discriminant $31.36$
Ramified primes $5, 7$
Class number $2$ (GRH)
Class group $[2]$ (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, 1960, 2576, -1498, -3171, -287, 1274, 471, 7, -126, -21, 7, 14, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 14*x^12 + 7*x^11 - 21*x^10 - 126*x^9 + 7*x^8 + 471*x^7 + 1274*x^6 - 287*x^5 - 3171*x^4 - 1498*x^3 + 2576*x^2 + 1960*x + 400)
 
gp: K = bnfinit(x^14 - 7*x^13 + 14*x^12 + 7*x^11 - 21*x^10 - 126*x^9 + 7*x^8 + 471*x^7 + 1274*x^6 - 287*x^5 - 3171*x^4 - 1498*x^3 + 2576*x^2 + 1960*x + 400, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 14 x^{12} + 7 x^{11} - 21 x^{10} - 126 x^{9} + 7 x^{8} + 471 x^{7} + 1274 x^{6} - 287 x^{5} - 3171 x^{4} - 1498 x^{3} + 2576 x^{2} + 1960 x + 400 \)

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:  \(-890538686357276796875=-\,5^{7}\cdot 7^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.36$
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:  $5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{20} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{20} a^{2} + \frac{1}{5} a$, $\frac{1}{20} a^{9} - \frac{1}{10} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a$, $\frac{1}{40} a^{10} - \frac{1}{40} a^{9} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{2}{5} a^{2} + \frac{1}{10} a$, $\frac{1}{40} a^{11} - \frac{1}{40} a^{9} + \frac{1}{5} a^{7} + \frac{7}{40} a^{5} + \frac{1}{40} a^{3} - \frac{2}{5} a$, $\frac{1}{400} a^{12} + \frac{1}{100} a^{11} + \frac{1}{100} a^{10} - \frac{1}{80} a^{9} + \frac{1}{50} a^{8} - \frac{7}{100} a^{7} - \frac{93}{400} a^{6} + \frac{1}{5} a^{5} + \frac{1}{25} a^{4} - \frac{71}{400} a^{3} - \frac{39}{100} a^{2} + \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{37442316220000} a^{13} - \frac{16349567729}{37442316220000} a^{12} + \frac{48427870463}{9360579055000} a^{11} - \frac{11438195137}{37442316220000} a^{10} + \frac{19534464451}{870751540000} a^{9} - \frac{43890792367}{2340144763750} a^{8} - \frac{806619940409}{37442316220000} a^{7} - \frac{5871308519231}{37442316220000} a^{6} - \frac{52276444927}{217687885000} a^{5} - \frac{8369262121719}{37442316220000} a^{4} - \frac{14410743194553}{37442316220000} a^{3} - \frac{1434633870779}{4680289527500} a^{2} + \frac{128498821601}{468028952750} a + \frac{8331781941}{187211581100}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 385659.501597 \) (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

\(\Q(\sqrt{-35}) \), 7.1.5044200875.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.5044200875.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
7Data not computed