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Properties

Label 14.0.10700490781...6387.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,23^{7}\cdot 61^{7}$
Root discriminant $37.46$
Ramified primes $23, 61$
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![38525, 26680, 25341, 18934, 9674, 4432, 2384, 386, 194, 114, 14, -16, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 5*x^12 - 16*x^11 + 14*x^10 + 114*x^9 + 194*x^8 + 386*x^7 + 2384*x^6 + 4432*x^5 + 9674*x^4 + 18934*x^3 + 25341*x^2 + 26680*x + 38525)
 
gp: K = bnfinit(x^14 - 2*x^13 - 5*x^12 - 16*x^11 + 14*x^10 + 114*x^9 + 194*x^8 + 386*x^7 + 2384*x^6 + 4432*x^5 + 9674*x^4 + 18934*x^3 + 25341*x^2 + 26680*x + 38525, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 5 x^{12} - 16 x^{11} + 14 x^{10} + 114 x^{9} + 194 x^{8} + 386 x^{7} + 2384 x^{6} + 4432 x^{5} + 9674 x^{4} + 18934 x^{3} + 25341 x^{2} + 26680 x + 38525 \)

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:  \(-10700490781461249026387=-\,23^{7}\cdot 61^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.46$
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:  $23, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} + \frac{1}{16} a^{6} + \frac{3}{16} a^{3} + \frac{3}{16}$, $\frac{1}{80} a^{10} + \frac{1}{80} a^{9} - \frac{1}{20} a^{8} + \frac{9}{80} a^{7} - \frac{7}{80} a^{6} - \frac{1}{5} a^{5} + \frac{11}{80} a^{4} + \frac{3}{80} a^{3} - \frac{7}{20} a^{2} - \frac{29}{80} a - \frac{1}{16}$, $\frac{1}{80} a^{11} - \frac{7}{80} a^{8} + \frac{1}{20} a^{7} - \frac{1}{20} a^{6} - \frac{13}{80} a^{5} - \frac{1}{10} a^{4} - \frac{1}{5} a^{3} - \frac{21}{80} a^{2} + \frac{1}{20} a + \frac{1}{4}$, $\frac{1}{7360} a^{12} - \frac{3}{920} a^{11} + \frac{17}{3680} a^{10} - \frac{17}{1840} a^{9} + \frac{77}{920} a^{8} + \frac{293}{3680} a^{7} + \frac{35}{736} a^{6} + \frac{11}{184} a^{5} - \frac{153}{736} a^{4} + \frac{89}{368} a^{3} + \frac{147}{920} a^{2} + \frac{73}{160} a + \frac{11}{64}$, $\frac{1}{42875414834567680} a^{13} - \frac{756450515503}{42875414834567680} a^{12} + \frac{56185775661503}{21437707417283840} a^{11} - \frac{842331946329}{186414847106816} a^{10} + \frac{304294107991963}{10718853708641920} a^{9} - \frac{1112526773973013}{21437707417283840} a^{8} + \frac{661824722457529}{10718853708641920} a^{7} - \frac{2632155463884281}{21437707417283840} a^{6} - \frac{1398725665429411}{21437707417283840} a^{5} + \frac{5044690136921903}{21437707417283840} a^{4} + \frac{310055952975593}{10718853708641920} a^{3} + \frac{9986108411039393}{21437707417283840} a^{2} + \frac{68227715237717}{1864148471068160} a - \frac{2073312424167}{5564622301696}$

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:  \( 517767.664818 \) (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

\(\Q(\sqrt{-1403}) \), 7.1.2761677827.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.2761677827.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
$61$61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$