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Properties

Label 14.14.1064726745...9969.1
Degree $14$
Signature $[14, 0]$
Discriminant $1009^{7}$
Root discriminant $31.76$
Ramified prime $1009$
Class number $1$
Class group Trivial
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![1, 70, 266, -337, -1817, -620, 2396, 1365, -1144, -620, 271, 91, -28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 28*x^12 + 91*x^11 + 271*x^10 - 620*x^9 - 1144*x^8 + 1365*x^7 + 2396*x^6 - 620*x^5 - 1817*x^4 - 337*x^3 + 266*x^2 + 70*x + 1)
 
gp: K = bnfinit(x^14 - 4*x^13 - 28*x^12 + 91*x^11 + 271*x^10 - 620*x^9 - 1144*x^8 + 1365*x^7 + 2396*x^6 - 620*x^5 - 1817*x^4 - 337*x^3 + 266*x^2 + 70*x + 1, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 28 x^{12} + 91 x^{11} + 271 x^{10} - 620 x^{9} - 1144 x^{8} + 1365 x^{7} + 2396 x^{6} - 620 x^{5} - 1817 x^{4} - 337 x^{3} + 266 x^{2} + 70 x + 1 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1064726745878753869969=1009^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.76$
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:  $1009$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19} a^{12} - \frac{4}{19} a^{11} + \frac{5}{19} a^{10} - \frac{3}{19} a^{9} - \frac{1}{19} a^{8} + \frac{3}{19} a^{7} + \frac{1}{19} a^{6} + \frac{1}{19} a^{5} - \frac{3}{19} a^{4} + \frac{2}{19} a^{3} + \frac{3}{19} a^{2} - \frac{5}{19} a + \frac{4}{19}$, $\frac{1}{18362946359437} a^{13} - \frac{401846150664}{18362946359437} a^{12} - \frac{6075282001740}{18362946359437} a^{11} - \frac{8413085763126}{18362946359437} a^{10} - \frac{6765069100713}{18362946359437} a^{9} + \frac{469662807714}{1080173315261} a^{8} + \frac{185442992758}{18362946359437} a^{7} + \frac{8565992257875}{18362946359437} a^{6} + \frac{1802188123608}{18362946359437} a^{5} + \frac{186889961862}{1080173315261} a^{4} + \frac{26835949711}{18362946359437} a^{3} + \frac{5304404920560}{18362946359437} a^{2} + \frac{275206976754}{966470861023} a + \frac{7100317428992}{18362946359437}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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:  \( 664881.724002 \)
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

\(\Q(\sqrt{1009}) \), 7.7.1027243729.1 x7

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

Sibling fields

Degree 7 sibling: 7.7.1027243729.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
1009Data not computed