↑

Properties

Label 14.0.91220553419...3487.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7103^{7}$
Root discriminant $84.28$
Ramified prime $7103$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
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![6141952, 2476800, -488384, -930864, 149428, 47320, -31723, -14660, 4008, 2492, 98, -28, -20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 20*x^12 - 28*x^11 + 98*x^10 + 2492*x^9 + 4008*x^8 - 14660*x^7 - 31723*x^6 + 47320*x^5 + 149428*x^4 - 930864*x^3 - 488384*x^2 + 2476800*x + 6141952)
 
gp: K = bnfinit(x^14 - 4*x^13 - 20*x^12 - 28*x^11 + 98*x^10 + 2492*x^9 + 4008*x^8 - 14660*x^7 - 31723*x^6 + 47320*x^5 + 149428*x^4 - 930864*x^3 - 488384*x^2 + 2476800*x + 6141952, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 20 x^{12} - 28 x^{11} + 98 x^{10} + 2492 x^{9} + 4008 x^{8} - 14660 x^{7} - 31723 x^{6} + 47320 x^{5} + 149428 x^{4} - 930864 x^{3} - 488384 x^{2} + 2476800 x + 6141952 \)

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:  \(-912205534197333179463463487=-\,7103^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.28$
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:  $7103$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{7} - \frac{1}{32} a^{6} - \frac{1}{32} a^{5} - \frac{1}{16} a^{4} + \frac{9}{64} a^{3} + \frac{3}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{128} a^{8} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} - \frac{7}{128} a^{4} + \frac{3}{16} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{128} a^{9} - \frac{1}{32} a^{6} - \frac{3}{128} a^{5} - \frac{7}{64} a^{3} + \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{2560} a^{10} - \frac{7}{2560} a^{9} + \frac{3}{1280} a^{8} + \frac{9}{1280} a^{7} + \frac{5}{512} a^{6} - \frac{87}{2560} a^{5} + \frac{1}{32} a^{4} - \frac{77}{640} a^{3} + \frac{33}{160} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{10240} a^{11} - \frac{3}{10240} a^{9} + \frac{1}{512} a^{8} + \frac{31}{10240} a^{7} - \frac{29}{1280} a^{6} + \frac{231}{10240} a^{5} + \frac{93}{2560} a^{4} + \frac{623}{2560} a^{3} + \frac{27}{320} a^{2} - \frac{3}{160} a + \frac{7}{20}$, $\frac{1}{10240} a^{12} + \frac{1}{10240} a^{10} - \frac{1}{1280} a^{9} - \frac{5}{2048} a^{8} - \frac{149}{10240} a^{6} + \frac{43}{1280} a^{5} + \frac{43}{2560} a^{4} - \frac{53}{640} a^{3} - \frac{7}{20} a + \frac{1}{5}$, $\frac{1}{7933426858698212392960} a^{13} + \frac{119780399974221579}{7933426858698212392960} a^{12} + \frac{65129855678015441}{7933426858698212392960} a^{11} - \frac{137489367541190041}{1586685371739642478592} a^{10} - \frac{1240521266585525437}{1586685371739642478592} a^{9} + \frac{8533213692972206269}{7933426858698212392960} a^{8} - \frac{6482970700634371717}{7933426858698212392960} a^{7} + \frac{5343581879932337797}{273566443403386634240} a^{6} + \frac{893175710283489361}{396671342934910619648} a^{5} + \frac{10264913096115316709}{396671342934910619648} a^{4} - \frac{119148030790172364079}{495839178668638274560} a^{3} + \frac{28837949134412570597}{123959794667159568640} a^{2} + \frac{14364773441775030939}{30989948666789892160} a + \frac{23873760690953915}{774748716669747304}$

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

Class group and class number

$C_{11}$, which has order $11$ (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:  \( 36266671461.4 \) (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

\(\Q(\sqrt{-7103}) \), 7.1.358364881727.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.358364881727.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
7103Data not computed