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Properties

Label 14.0.49098849571...2359.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,4679^{7}$
Root discriminant $68.40$
Ramified prime $4679$
Class number $13$ (GRH)
Class group $[13]$ (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![3910177, 5433177, 794668, -1255412, -167580, 192779, 48948, -18667, -3663, 595, 369, -21, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 6*x^12 - 21*x^11 + 369*x^10 + 595*x^9 - 3663*x^8 - 18667*x^7 + 48948*x^6 + 192779*x^5 - 167580*x^4 - 1255412*x^3 + 794668*x^2 + 5433177*x + 3910177)
 
gp: K = bnfinit(x^14 - 4*x^13 - 6*x^12 - 21*x^11 + 369*x^10 + 595*x^9 - 3663*x^8 - 18667*x^7 + 48948*x^6 + 192779*x^5 - 167580*x^4 - 1255412*x^3 + 794668*x^2 + 5433177*x + 3910177, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 6 x^{12} - 21 x^{11} + 369 x^{10} + 595 x^{9} - 3663 x^{8} - 18667 x^{7} + 48948 x^{6} + 192779 x^{5} - 167580 x^{4} - 1255412 x^{3} + 794668 x^{2} + 5433177 x + 3910177 \)

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:  \(-49098849571309452996402359=-\,4679^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.40$
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:  $4679$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{103} a^{12} + \frac{24}{103} a^{11} - \frac{45}{103} a^{10} - \frac{11}{103} a^{9} + \frac{23}{103} a^{8} - \frac{4}{103} a^{7} - \frac{43}{103} a^{6} - \frac{32}{103} a^{5} + \frac{36}{103} a^{4} + \frac{33}{103} a^{3} + \frac{49}{103} a^{2} + \frac{6}{103} a - \frac{40}{103}$, $\frac{1}{33284030892872082119360274959862658697} a^{13} - \frac{58397750651197915763298132705683651}{33284030892872082119360274959862658697} a^{12} + \frac{13473356352461330518564256007515549742}{33284030892872082119360274959862658697} a^{11} + \frac{4563708843054927852895782242253749926}{33284030892872082119360274959862658697} a^{10} - \frac{23662602773202354586921033986024559}{117611416582586862612580476889974059} a^{9} + \frac{14045299528090638760592905875567217285}{33284030892872082119360274959862658697} a^{8} + \frac{14229687820876901703248160681927143202}{33284030892872082119360274959862658697} a^{7} + \frac{1476541288292584257306231052545499404}{33284030892872082119360274959862658697} a^{6} + \frac{10051781690600625441722610956119095489}{33284030892872082119360274959862658697} a^{5} + \frac{10174869475210629153450015398643404072}{33284030892872082119360274959862658697} a^{4} - \frac{257440289026261902187819846224107609}{774047230066792607426983138601457179} a^{3} + \frac{3131217946067722498326478681983766558}{33284030892872082119360274959862658697} a^{2} + \frac{9349678529591261769460262317597533943}{33284030892872082119360274959862658697} a + \frac{10263680093571381127269696022857314513}{33284030892872082119360274959862658697}$

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

Class group and class number

$C_{13}$, which has order $13$ (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:  \( 419390.665529 \) (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

\(\Q(\sqrt{-4679}) \), 7.1.102437538839.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.102437538839.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
4679Data not computed