↑

Properties

Label 14.0.13971315557...1831.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,17^{7}\cdot 23^{7}$
Root discriminant $19.77$
Ramified primes $17, 23$
Class number $2$
Class group $[2]$
Galois group $D_{7}$ (as 14T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![391, 391, 995, 342, 421, 188, -136, 78, -28, -76, 28, 15, -7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 7*x^12 + 15*x^11 + 28*x^10 - 76*x^9 - 28*x^8 + 78*x^7 - 136*x^6 + 188*x^5 + 421*x^4 + 342*x^3 + 995*x^2 + 391*x + 391)
 
gp: K = bnfinit(x^14 - 2*x^13 - 7*x^12 + 15*x^11 + 28*x^10 - 76*x^9 - 28*x^8 + 78*x^7 - 136*x^6 + 188*x^5 + 421*x^4 + 342*x^3 + 995*x^2 + 391*x + 391, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 7 x^{12} + 15 x^{11} + 28 x^{10} - 76 x^{9} - 28 x^{8} + 78 x^{7} - 136 x^{6} + 188 x^{5} + 421 x^{4} + 342 x^{3} + 995 x^{2} + 391 x + 391 \)

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:  \(-1397131555718611831=-\,17^{7}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.77$
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:  $17, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{2}{9}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{4}{27} a^{9} - \frac{1}{27} a^{8} - \frac{2}{9} a^{7} + \frac{1}{9} a^{6} + \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{4}{27} a^{3} - \frac{7}{27} a^{2} + \frac{10}{27} a + \frac{10}{27}$, $\frac{1}{42687} a^{12} + \frac{19}{2511} a^{11} - \frac{371}{14229} a^{10} - \frac{6716}{42687} a^{9} + \frac{265}{2511} a^{8} - \frac{2524}{14229} a^{7} + \frac{5069}{14229} a^{6} - \frac{1762}{14229} a^{5} + \frac{44}{42687} a^{4} - \frac{20465}{42687} a^{3} - \frac{2071}{14229} a^{2} + \frac{136}{2511} a + \frac{380}{2511}$, $\frac{1}{1574771111379} a^{13} - \frac{976183}{174974567931} a^{12} - \frac{27450581776}{1574771111379} a^{11} + \frac{16577166430}{1574771111379} a^{10} + \frac{7492098859}{524923703793} a^{9} - \frac{199459098574}{1574771111379} a^{8} + \frac{119157270847}{524923703793} a^{7} + \frac{86999211238}{524923703793} a^{6} + \frac{336639791336}{1574771111379} a^{5} - \frac{24065704843}{58324855977} a^{4} - \frac{49064141486}{1574771111379} a^{3} - \frac{702106276279}{1574771111379} a^{2} + \frac{8823622066}{30877864929} a + \frac{35540504525}{92633594787}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1673.20138775 \)
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

\(\Q(\sqrt{-391}) \), 7.1.59776471.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.59776471.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$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$$[\ ]_{2}$
$23$23.2.1.1$x^{2} - 23$$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}$