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Properties

Label 14.0.26659081242...1559.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,17^{7}\cdot 487^{7}$
Root discriminant $90.99$
Ramified primes $17, 487$
Class number $18$ (GRH)
Class group $[18]$ (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![1413721, -531483, 1333379, -1497260, 872694, -540701, 301240, -102392, 18141, 21, -987, 228, -17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 17*x^12 + 228*x^11 - 987*x^10 + 21*x^9 + 18141*x^8 - 102392*x^7 + 301240*x^6 - 540701*x^5 + 872694*x^4 - 1497260*x^3 + 1333379*x^2 - 531483*x + 1413721)
 
gp: K = bnfinit(x^14 - 2*x^13 - 17*x^12 + 228*x^11 - 987*x^10 + 21*x^9 + 18141*x^8 - 102392*x^7 + 301240*x^6 - 540701*x^5 + 872694*x^4 - 1497260*x^3 + 1333379*x^2 - 531483*x + 1413721, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 17 x^{12} + 228 x^{11} - 987 x^{10} + 21 x^{9} + 18141 x^{8} - 102392 x^{7} + 301240 x^{6} - 540701 x^{5} + 872694 x^{4} - 1497260 x^{3} + 1333379 x^{2} - 531483 x + 1413721 \)

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:  \(-2665908124227273813779851559=-\,17^{7}\cdot 487^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.99$
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:  $17, 487$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{109} a^{11} + \frac{52}{109} a^{10} + \frac{35}{109} a^{9} + \frac{30}{109} a^{8} - \frac{25}{109} a^{7} + \frac{47}{109} a^{6} - \frac{20}{109} a^{5} - \frac{52}{109} a^{4} + \frac{38}{109} a^{3} + \frac{43}{109} a^{2} - \frac{6}{109} a - \frac{52}{109}$, $\frac{1}{2033569291} a^{12} - \frac{5013254}{2033569291} a^{11} - \frac{13416692}{119621723} a^{10} + \frac{226413069}{2033569291} a^{9} + \frac{49436814}{156428407} a^{8} - \frac{888400221}{2033569291} a^{7} - \frac{203544304}{2033569291} a^{6} + \frac{385990711}{2033569291} a^{5} + \frac{803699539}{2033569291} a^{4} - \frac{236144866}{2033569291} a^{3} + \frac{21847974}{70123079} a^{2} - \frac{198202404}{2033569291} a + \frac{829371}{1710319}$, $\frac{1}{21590916095518983525263836963} a^{13} - \frac{1879346059758540317}{21590916095518983525263836963} a^{12} - \frac{8178880976914607148596429}{21590916095518983525263836963} a^{11} - \frac{4988460896823108086136838367}{21590916095518983525263836963} a^{10} + \frac{971534463839676691493508889}{21590916095518983525263836963} a^{9} + \frac{79385951862383676457353155}{198081799041458564451961807} a^{8} - \frac{3373932454665574016426124261}{21590916095518983525263836963} a^{7} - \frac{6061949694640625152253822095}{21590916095518983525263836963} a^{6} + \frac{10792477979410107450931236866}{21590916095518983525263836963} a^{5} - \frac{6020737912457722864513999913}{21590916095518983525263836963} a^{4} + \frac{531109603288403918171258434}{1270053887971704913250813939} a^{3} + \frac{552872086149016402623956916}{21590916095518983525263836963} a^{2} - \frac{10606501131815549624166608101}{21590916095518983525263836963} a + \frac{2808485890545493547821969}{18158886539544981938825767}$

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

Class group and class number

$C_{18}$, which has order $18$ (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:  \( 3283388.80428 \) (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

\(\Q(\sqrt{-8279}) \), 7.1.567457901639.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.567457901639.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
487Data not computed