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Properties

Label 14.0.37169536547...9872.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 11^{7}\cdot 71^{7}$
Root discriminant $79.04$
Ramified primes $2, 11, 71$
Class number $676$ (GRH)
Class group $[26, 26]$ (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![17351728, -6183696, -2238836, 818984, 35788, 182464, -56679, -20468, 3928, 196, 666, -196, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 12*x^12 - 196*x^11 + 666*x^10 + 196*x^9 + 3928*x^8 - 20468*x^7 - 56679*x^6 + 182464*x^5 + 35788*x^4 + 818984*x^3 - 2238836*x^2 - 6183696*x + 17351728)
 
gp: K = bnfinit(x^14 - 4*x^13 + 12*x^12 - 196*x^11 + 666*x^10 + 196*x^9 + 3928*x^8 - 20468*x^7 - 56679*x^6 + 182464*x^5 + 35788*x^4 + 818984*x^3 - 2238836*x^2 - 6183696*x + 17351728, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 12 x^{12} - 196 x^{11} + 666 x^{10} + 196 x^{9} + 3928 x^{8} - 20468 x^{7} - 56679 x^{6} + 182464 x^{5} + 35788 x^{4} + 818984 x^{3} - 2238836 x^{2} - 6183696 x + 17351728 \)

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:  \(-371695365478652317502799872=-\,2^{21}\cdot 11^{7}\cdot 71^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.04$
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:  $2, 11, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{88} a^{8} - \frac{3}{44} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{19}{44} a^{2} - \frac{4}{11}$, $\frac{1}{88} a^{9} - \frac{3}{44} a^{7} + \frac{1}{8} a^{5} + \frac{19}{44} a^{3} - \frac{1}{2} a^{2} - \frac{4}{11} a$, $\frac{1}{440} a^{10} - \frac{1}{440} a^{9} - \frac{1}{440} a^{8} - \frac{19}{220} a^{7} + \frac{47}{440} a^{6} + \frac{1}{8} a^{5} - \frac{61}{440} a^{4} + \frac{7}{110} a^{3} - \frac{9}{220} a^{2} + \frac{41}{110} a + \frac{24}{55}$, $\frac{1}{880} a^{11} + \frac{3}{880} a^{9} - \frac{1}{220} a^{8} - \frac{21}{880} a^{7} - \frac{27}{220} a^{6} + \frac{49}{880} a^{5} + \frac{3}{20} a^{4} - \frac{1}{44} a^{3} + \frac{39}{220} a^{2} - \frac{61}{220} a + \frac{49}{110}$, $\frac{1}{96800} a^{12} - \frac{1}{4840} a^{11} + \frac{3}{19360} a^{10} - \frac{73}{48400} a^{9} + \frac{447}{96800} a^{8} + \frac{997}{12100} a^{7} - \frac{4687}{96800} a^{6} - \frac{8509}{48400} a^{5} - \frac{3653}{24200} a^{4} - \frac{3411}{12100} a^{3} - \frac{29}{968} a^{2} - \frac{313}{2420} a + \frac{1462}{3025}$, $\frac{1}{192112113029753820761580800} a^{13} + \frac{744872754130580165309}{192112113029753820761580800} a^{12} - \frac{10004279871640755938123}{38422422605950764152316160} a^{11} + \frac{111681915874941306631909}{192112113029753820761580800} a^{10} - \frac{931687437703593091314657}{192112113029753820761580800} a^{9} - \frac{790336258242711848996381}{192112113029753820761580800} a^{8} + \frac{19892276917140218115869307}{192112113029753820761580800} a^{7} - \frac{11045386399165721761376601}{192112113029753820761580800} a^{6} + \frac{1453252760814614532571613}{6003503532179806898799400} a^{5} - \frac{1240161624582967818474483}{6003503532179806898799400} a^{4} - \frac{307251216537059407999477}{980163841988539901844800} a^{3} + \frac{4600959097646854775915441}{9605605651487691038079040} a^{2} - \frac{1632179373148923181795813}{6003503532179806898799400} a + \frac{665688810562035470914061}{12007007064359613797598800}$

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

Class group and class number

$C_{26}\times C_{26}$, which has order $676$ (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:  \( 6726511.81525 \) (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

\(\Q(\sqrt{-1562}) \), 7.1.243906324992.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.243906324992.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$2$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
$71$71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$