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Properties

Label 14.0.32918715676...4443.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 43^{13}$
Root discriminant $923.70$
Ramified primes $7, 43$
Class number $387499$ (GRH)
Class group $[387499]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1580168235629, -431820674782, 610422315916, -87810748963, 8912217496, -1029856751, 1161357330, 133073863, 23613149, 780794, 43344, -5719, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5719*x^11 + 43344*x^10 + 780794*x^9 + 23613149*x^8 + 133073863*x^7 + 1161357330*x^6 - 1029856751*x^5 + 8912217496*x^4 - 87810748963*x^3 + 610422315916*x^2 - 431820674782*x + 1580168235629)
 
gp: K = bnfinit(x^14 - 5719*x^11 + 43344*x^10 + 780794*x^9 + 23613149*x^8 + 133073863*x^7 + 1161357330*x^6 - 1029856751*x^5 + 8912217496*x^4 - 87810748963*x^3 + 610422315916*x^2 - 431820674782*x + 1580168235629, 1)
 

Normalized defining polynomial

\( x^{14} - 5719 x^{11} + 43344 x^{10} + 780794 x^{9} + 23613149 x^{8} + 133073863 x^{7} + 1161357330 x^{6} - 1029856751 x^{5} + 8912217496 x^{4} - 87810748963 x^{3} + 610422315916 x^{2} - 431820674782 x + 1580168235629 \)

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:  \(-329187156767051876908796215815371233484443=-\,7^{24}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $923.70$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2107=7^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2107}(1408,·)$, $\chi_{2107}(1,·)$, $\chi_{2107}(610,·)$, $\chi_{2107}(211,·)$, $\chi_{2107}(715,·)$, $\chi_{2107}(2066,·)$, $\chi_{2107}(687,·)$, $\chi_{2107}(1681,·)$, $\chi_{2107}(274,·)$, $\chi_{2107}(1331,·)$, $\chi_{2107}(1268,·)$, $\chi_{2107}(183,·)$, $\chi_{2107}(1884,·)$, $\chi_{2107}(925,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{27} a^{12} + \frac{2}{27} a^{11} - \frac{1}{27} a^{10} + \frac{2}{27} a^{9} - \frac{2}{27} a^{7} - \frac{11}{27} a^{6} + \frac{13}{27} a^{5} - \frac{2}{9} a^{4} - \frac{4}{27} a^{3} - \frac{10}{27} a^{2} + \frac{11}{27} a + \frac{13}{27}$, $\frac{1}{105516435199243382264019480285594396608350139697970216004897900291709603579} a^{13} - \frac{29590128809310479407177146677180755151168634910860918749192591791557721}{35172145066414460754673160095198132202783379899323405334965966763903201193} a^{12} - \frac{10823848695302125097795318729779985137536681562558248030057645250089343682}{105516435199243382264019480285594396608350139697970216004897900291709603579} a^{11} - \frac{131158558671750873128911557882385212734445680730690128022320198374140421}{2245030536154114516255733623097753119326598716978089702231870218972544757} a^{10} - \frac{7804001779982293416520545357105132243092943366825706074209189495296198450}{105516435199243382264019480285594396608350139697970216004897900291709603579} a^{9} + \frac{152390277004562911753515590895654870917402545915308612577227295177102951}{1335651078471435218531892155513853121624685312632534379808834180907716501} a^{8} + \frac{5265806659778974139071152863052277413600780509990983446191053048770115382}{105516435199243382264019480285594396608350139697970216004897900291709603579} a^{7} - \frac{47138749587955976384550892104227685981859171606034263877787160588666740640}{105516435199243382264019480285594396608350139697970216004897900291709603579} a^{6} + \frac{18736399944145262017758798244462654883449404517153485299621286975032223163}{105516435199243382264019480285594396608350139697970216004897900291709603579} a^{5} - \frac{14447113989673259046636322106667103930088130381496877169649514470987585618}{105516435199243382264019480285594396608350139697970216004897900291709603579} a^{4} + \frac{23325858099082815318241526014114845220945050650723358228073546768668034459}{105516435199243382264019480285594396608350139697970216004897900291709603579} a^{3} - \frac{31477466820352817575173386106987699183137548948540213620334769085106060973}{105516435199243382264019480285594396608350139697970216004897900291709603579} a^{2} + \frac{2810859873275463278536321324986512301785148945755362347978690431635338391}{35172145066414460754673160095198132202783379899323405334965966763903201193} a - \frac{30616442051452442122708341746728194294932760418424154482705730738813137873}{105516435199243382264019480285594396608350139697970216004897900291709603579}$

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

Class group and class number

$C_{387499}$, which has order $387499$ (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:  \( 5135341547.611097 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-43}) \), 7.7.87495801462998035849.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
$43$43.14.13.13$x^{14} + 1349385563187$$14$$1$$13$$C_{14}$$[\ ]_{14}$