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Properties

Label 14.2.18709104627...5625.1
Degree $14$
Signature $[2, 6]$
Discriminant $5^{10}\cdot 7^{24}$
Root discriminant $88.72$
Ramified primes $5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:D_7.C_2$ (as 14T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-299295, 265545, -383670, 627165, -117859, 63049, 17297, -10485, 8869, -2282, 1008, -112, 49, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 49*x^12 - 112*x^11 + 1008*x^10 - 2282*x^9 + 8869*x^8 - 10485*x^7 + 17297*x^6 + 63049*x^5 - 117859*x^4 + 627165*x^3 - 383670*x^2 + 265545*x - 299295)
 
gp: K = bnfinit(x^14 + 49*x^12 - 112*x^11 + 1008*x^10 - 2282*x^9 + 8869*x^8 - 10485*x^7 + 17297*x^6 + 63049*x^5 - 117859*x^4 + 627165*x^3 - 383670*x^2 + 265545*x - 299295, 1)
 

Normalized defining polynomial

\( x^{14} + 49 x^{12} - 112 x^{11} + 1008 x^{10} - 2282 x^{9} + 8869 x^{8} - 10485 x^{7} + 17297 x^{6} + 63049 x^{5} - 117859 x^{4} + 627165 x^{3} - 383670 x^{2} + 265545 x - 299295 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1870910462700843890634765625=5^{10}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.72$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{12} a^{10} + \frac{1}{6} a^{9} + \frac{5}{12} a^{8} - \frac{1}{4} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{3} a^{2} + \frac{5}{12} a + \frac{1}{4}$, $\frac{1}{72} a^{11} - \frac{1}{72} a^{10} - \frac{1}{72} a^{9} - \frac{5}{24} a^{8} + \frac{7}{24} a^{7} + \frac{1}{72} a^{6} - \frac{1}{12} a^{5} - \frac{1}{8} a^{4} + \frac{17}{72} a^{3} + \frac{5}{72} a^{2} + \frac{1}{3} a + \frac{3}{8}$, $\frac{1}{144} a^{12} - \frac{1}{72} a^{10} - \frac{1}{9} a^{9} + \frac{1}{24} a^{8} + \frac{11}{72} a^{7} + \frac{67}{144} a^{6} + \frac{19}{48} a^{5} + \frac{1}{18} a^{4} - \frac{25}{72} a^{3} - \frac{43}{144} a^{2} + \frac{17}{48} a + \frac{3}{16}$, $\frac{1}{1900673143998405775793212778451648} a^{13} + \frac{1458839339384810444782903044529}{633557714666135258597737592817216} a^{12} + \frac{1342593881899575372266032991861}{950336571999202887896606389225824} a^{11} + \frac{29571863377275338408607466937743}{950336571999202887896606389225824} a^{10} + \frac{35110385458670742736795478686799}{316778857333067629298868796408608} a^{9} + \frac{173098157857736301073681042804453}{475168285999601443948303194612912} a^{8} - \frac{720021496378066874792456369172815}{1900673143998405775793212778451648} a^{7} + \frac{14195453072906340777742691014645}{316778857333067629298868796408608} a^{6} + \frac{687745927764772391150712582440651}{1900673143998405775793212778451648} a^{5} - \frac{39651670366317059158686788312539}{950336571999202887896606389225824} a^{4} + \frac{254267135992939602463798346024747}{1900673143998405775793212778451648} a^{3} - \frac{74017544928433700714647680723739}{316778857333067629298868796408608} a^{2} - \frac{5261319316540720956774007599373}{17598825407392646072159377578256} a + \frac{29428862151946521071510759662815}{70395301629570584288637510313024}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $7$
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:  \( 336700643.52 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:D_7.C_2$ (as 14T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 196
The 16 conjugacy class representatives for $C_7:D_7.C_2$
Character table for $C_7:D_7.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \)

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

Sibling fields

Degree 14 siblings: data not computed
Degree 28 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
7Data not computed