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Properties

Label 14.0.28033829543...9839.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 617^{7}$
Root discriminant $65.72$
Ramified primes $7, 617$
Class number $12$ (GRH)
Class group $[12]$ (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![1477683, -2208519, 960957, 15948, 250397, -126275, 52503, -11168, 1841, 63, -86, -10, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 12*x^12 - 10*x^11 - 86*x^10 + 63*x^9 + 1841*x^8 - 11168*x^7 + 52503*x^6 - 126275*x^5 + 250397*x^4 + 15948*x^3 + 960957*x^2 - 2208519*x + 1477683)
 
gp: K = bnfinit(x^14 - 4*x^13 + 12*x^12 - 10*x^11 - 86*x^10 + 63*x^9 + 1841*x^8 - 11168*x^7 + 52503*x^6 - 126275*x^5 + 250397*x^4 + 15948*x^3 + 960957*x^2 - 2208519*x + 1477683, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 12 x^{12} - 10 x^{11} - 86 x^{10} + 63 x^{9} + 1841 x^{8} - 11168 x^{7} + 52503 x^{6} - 126275 x^{5} + 250397 x^{4} + 15948 x^{3} + 960957 x^{2} - 2208519 x + 1477683 \)

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:  \(-28033829543340009100599839=-\,7^{7}\cdot 617^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.72$
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:  $7, 617$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{8} + \frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{27} a^{2} - \frac{2}{9} a$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{1701} a^{10} + \frac{5}{1701} a^{9} + \frac{1}{567} a^{8} + \frac{22}{567} a^{7} + \frac{25}{567} a^{6} - \frac{16}{567} a^{5} + \frac{152}{1701} a^{4} - \frac{221}{1701} a^{3} - \frac{104}{567} a^{2} - \frac{50}{189} a + \frac{3}{7}$, $\frac{1}{15309} a^{11} - \frac{2}{15309} a^{10} - \frac{32}{15309} a^{9} - \frac{2}{1701} a^{8} - \frac{64}{1701} a^{7} - \frac{2}{5103} a^{6} - \frac{457}{15309} a^{5} + \frac{1739}{15309} a^{4} + \frac{1991}{15309} a^{3} + \frac{536}{5103} a^{2} - \frac{661}{1701} a + \frac{1}{9}$, $\frac{1}{2893401} a^{12} - \frac{8}{964467} a^{11} + \frac{1}{19683} a^{10} + \frac{5330}{2893401} a^{9} + \frac{2356}{321489} a^{8} + \frac{43480}{964467} a^{7} - \frac{9049}{413343} a^{6} - \frac{6227}{137781} a^{5} - \frac{113546}{964467} a^{4} + \frac{148534}{2893401} a^{3} - \frac{478769}{964467} a^{2} - \frac{15644}{321489} a - \frac{1639}{11907}$, $\frac{1}{606461746745544631641} a^{13} - \frac{2416868883001}{606461746745544631641} a^{12} + \frac{223273669300631}{22461546175760912283} a^{11} + \frac{74357614495712789}{606461746745544631641} a^{10} - \frac{1659441539854310570}{606461746745544631641} a^{9} - \frac{2967805112501500913}{202153915581848210547} a^{8} - \frac{27914209074431552830}{606461746745544631641} a^{7} + \frac{959101357120857280}{86637392392220661663} a^{6} - \frac{3330467942912261839}{67384638527282736849} a^{5} - \frac{97199222917869401477}{606461746745544631641} a^{4} + \frac{98588235344799257462}{606461746745544631641} a^{3} - \frac{68308044265294134286}{202153915581848210547} a^{2} + \frac{14851545230317655123}{67384638527282736849} a + \frac{394957617688841098}{2495727352862323587}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 211347808.233 \) (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

\(\Q(\sqrt{-4319}) \), 7.1.80565593759.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.80565593759.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
617Data not computed