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Properties

Label 14.0.13760207184...6523.3
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{13}$
Root discriminant $736.28$
Ramified primes $7, 29$
Class number $11598916$ (GRH)
Class group $[11598916]$ (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![157023868437, -301664396124, 220567618782, -17074022049, 661193736, 952727923, -22304016, 25631157, 872291, -116928, 7308, -3451, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3451*x^11 + 7308*x^10 - 116928*x^9 + 872291*x^8 + 25631157*x^7 - 22304016*x^6 + 952727923*x^5 + 661193736*x^4 - 17074022049*x^3 + 220567618782*x^2 - 301664396124*x + 157023868437)
 
gp: K = bnfinit(x^14 - 3451*x^11 + 7308*x^10 - 116928*x^9 + 872291*x^8 + 25631157*x^7 - 22304016*x^6 + 952727923*x^5 + 661193736*x^4 - 17074022049*x^3 + 220567618782*x^2 - 301664396124*x + 157023868437, 1)
 

Normalized defining polynomial

\( x^{14} - 3451 x^{11} + 7308 x^{10} - 116928 x^{9} + 872291 x^{8} + 25631157 x^{7} - 22304016 x^{6} + 952727923 x^{5} + 661193736 x^{4} - 17074022049 x^{3} + 220567618782 x^{2} - 301664396124 x + 157023868437 \)

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:  \(-13760207184971837367389621368916758066523=-\,7^{25}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $736.28$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(1280,·)$, $\chi_{1421}(1,·)$, $\chi_{1421}(1252,·)$, $\chi_{1421}(645,·)$, $\chi_{1421}(328,·)$, $\chi_{1421}(169,·)$, $\chi_{1421}(1420,·)$, $\chi_{1421}(13,·)$, $\chi_{1421}(776,·)$, $\chi_{1421}(141,·)$, $\chi_{1421}(1009,·)$, $\chi_{1421}(1408,·)$, $\chi_{1421}(412,·)$, $\chi_{1421}(1093,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{27} a^{6} + \frac{1}{27} a^{4} + \frac{1}{9} a^{3} + \frac{7}{27} a^{2} - \frac{4}{9} a$, $\frac{1}{27} a^{7} + \frac{1}{27} a^{5} + \frac{1}{9} a^{4} - \frac{2}{27} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{243} a^{8} - \frac{1}{243} a^{7} + \frac{1}{243} a^{6} + \frac{11}{243} a^{5} - \frac{5}{243} a^{4} + \frac{26}{243} a^{3} - \frac{29}{81} a^{2} + \frac{2}{9} a$, $\frac{1}{243} a^{9} + \frac{1}{81} a^{6} + \frac{2}{81} a^{5} + \frac{4}{81} a^{4} - \frac{7}{243} a^{3} - \frac{32}{81} a^{2} + \frac{1}{3} a$, $\frac{1}{243} a^{10} + \frac{1}{81} a^{7} - \frac{1}{81} a^{6} + \frac{4}{81} a^{5} - \frac{16}{243} a^{4} + \frac{13}{81} a^{3} + \frac{2}{27} a^{2} - \frac{2}{9} a$, $\frac{1}{6561} a^{11} - \frac{7}{6561} a^{10} - \frac{2}{2187} a^{9} + \frac{1}{729} a^{8} - \frac{25}{2187} a^{7} + \frac{22}{2187} a^{6} - \frac{763}{6561} a^{5} + \frac{931}{6561} a^{4} - \frac{133}{2187} a^{3} - \frac{2}{9} a^{2} + \frac{7}{27} a$, $\frac{1}{72171} a^{12} + \frac{5}{72171} a^{11} - \frac{13}{8019} a^{10} + \frac{8}{8019} a^{9} + \frac{20}{24057} a^{8} + \frac{23}{2187} a^{7} - \frac{1159}{72171} a^{6} + \frac{3007}{72171} a^{5} + \frac{272}{2673} a^{4} - \frac{82}{8019} a^{3} - \frac{92}{297} a^{2} + \frac{1}{11} a + \frac{3}{11}$, $\frac{1}{4330751329419652045942589304419313031414522784733303} a^{13} - \frac{23819874508056786869304595081386563717115720188}{4330751329419652045942589304419313031414522784733303} a^{12} - \frac{110633985075727002146050388392472766925106823239}{4330751329419652045942589304419313031414522784733303} a^{11} - \frac{2131570562966044893304680813365766224785235222428}{1443583776473217348647529768139771010471507594911101} a^{10} + \frac{1244585495737279249886277406715962623366420556721}{1443583776473217348647529768139771010471507594911101} a^{9} + \frac{2105729841113552400742154244844021041945647980517}{1443583776473217348647529768139771010471507594911101} a^{8} - \frac{49586957462344835066218642119496770702728560900333}{4330751329419652045942589304419313031414522784733303} a^{7} + \frac{3175219216828947122833731763718250585478217171252}{4330751329419652045942589304419313031414522784733303} a^{6} - \frac{348926782167093858389618132987030736816739950057928}{4330751329419652045942589304419313031414522784733303} a^{5} + \frac{224555644813881798052859646385627676199018547579849}{1443583776473217348647529768139771010471507594911101} a^{4} + \frac{9530526290784494423823598120158980178746722499414}{160398197385913038738614418682196778941278621656789} a^{3} + \frac{144042766083611991742634590955344296436611581323}{336264564750341800290596265581125322728047424857} a^{2} - \frac{699640944834071540468741618791081972796220768242}{1980224659085346157266844675088849122731834835269} a - \frac{702663419045907396981654669249285598680113738}{1560460724259532038823360658068439025005386001}$

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

Class group and class number

$C_{11598916}$, which has order $11598916$ (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:  \( 4115791439.997621 \) (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

\(\Q(\sqrt{-203}) \), 7.7.8233120419813614521.3

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.25.54$x^{14} - 119 x^{13} + 133 x^{12} - 98 x^{11} + 98 x^{10} + 98 x^{9} - 49 x^{8} + 21 x^{7} + 147 x^{6} + 98 x^{5} - 49 x^{4} + 49 x^{2} - 147 x + 112$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.14.13.5$x^{14} - 7424$$14$$1$$13$$C_{14}$$[\ ]_{14}$