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Properties

Label 14.0.13162729586...3152.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 251^{7}$
Root discriminant $44.81$
Ramified primes $2, 251$
Class number $338$ (GRH)
Class group $[13, 26]$ (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![148464, -281808, 292724, -134240, 38232, -9220, 2241, 1628, -56, -316, -22, -44, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 16*x^12 - 44*x^11 - 22*x^10 - 316*x^9 - 56*x^8 + 1628*x^7 + 2241*x^6 - 9220*x^5 + 38232*x^4 - 134240*x^3 + 292724*x^2 - 281808*x + 148464)
 
gp: K = bnfinit(x^14 + 16*x^12 - 44*x^11 - 22*x^10 - 316*x^9 - 56*x^8 + 1628*x^7 + 2241*x^6 - 9220*x^5 + 38232*x^4 - 134240*x^3 + 292724*x^2 - 281808*x + 148464, 1)
 

Normalized defining polynomial

\( x^{14} + 16 x^{12} - 44 x^{11} - 22 x^{10} - 316 x^{9} - 56 x^{8} + 1628 x^{7} + 2241 x^{6} - 9220 x^{5} + 38232 x^{4} - 134240 x^{3} + 292724 x^{2} - 281808 x + 148464 \)

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:  \(-131627295869636184113152=-\,2^{21}\cdot 251^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.81$
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:  $2, 251$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{24} a^{8} + \frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} - \frac{5}{24} a^{4} + \frac{1}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{6} a$, $\frac{1}{96} a^{9} - \frac{1}{96} a^{8} - \frac{1}{48} a^{7} + \frac{1}{48} a^{6} + \frac{7}{96} a^{5} - \frac{19}{96} a^{4} - \frac{19}{48} a^{3} - \frac{11}{48} a^{2} - \frac{1}{4}$, $\frac{1}{96} a^{10} + \frac{1}{96} a^{8} + \frac{1}{12} a^{7} + \frac{1}{96} a^{6} + \frac{5}{24} a^{5} + \frac{19}{96} a^{4} - \frac{1}{24} a^{3} + \frac{3}{16} a^{2} - \frac{5}{12} a - \frac{1}{4}$, $\frac{1}{96} a^{11} + \frac{1}{96} a^{8} + \frac{11}{96} a^{7} + \frac{5}{48} a^{6} + \frac{5}{24} a^{5} - \frac{17}{96} a^{4} - \frac{1}{12} a^{3} + \frac{23}{48} a^{2} + \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{480} a^{12} + \frac{1}{240} a^{10} + \frac{1}{480} a^{9} + \frac{1}{96} a^{8} + \frac{1}{48} a^{7} + \frac{7}{240} a^{6} - \frac{17}{480} a^{5} - \frac{5}{48} a^{4} - \frac{1}{240} a^{3} - \frac{19}{40} a^{2} - \frac{3}{20} a - \frac{3}{10}$, $\frac{1}{9927391652497601780160} a^{13} + \frac{2665887838196169711}{3309130550832533926720} a^{12} + \frac{33406381759620547}{9927391652497601780160} a^{11} - \frac{33431777049375805193}{9927391652497601780160} a^{10} + \frac{40308987769240382453}{9927391652497601780160} a^{9} - \frac{37821888372720548467}{1985478330499520356032} a^{8} - \frac{48873493946380775371}{9927391652497601780160} a^{7} + \frac{149923596847638311837}{1985478330499520356032} a^{6} + \frac{296183697170622111377}{4963695826248800890080} a^{5} - \frac{606951677571618262771}{4963695826248800890080} a^{4} - \frac{15243831127764578393}{47727844483161547020} a^{3} + \frac{22533003061968568163}{206820659427033370420} a^{2} + \frac{559046717234422567}{1590928149438718234} a - \frac{13694256686521807907}{103410329713516685210}$

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

Class group and class number

$C_{13}\times C_{26}$, which has order $338$ (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:  \( 248174.330546 \) (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

\(\Q(\sqrt{-502}) \), 7.1.8096384512.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.8096384512.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
251Data not computed