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Properties

Label 14.0.10477098591...6875.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 7^{25}$
Root discriminant $72.21$
Ramified primes $5, 7$
Class number $254$ (GRH)
Class group $[254]$ (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![30361, -4557, 2863, -58940, 88655, -61572, 36995, -15735, 6174, -2702, 553, -119, 42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 42*x^12 - 119*x^11 + 553*x^10 - 2702*x^9 + 6174*x^8 - 15735*x^7 + 36995*x^6 - 61572*x^5 + 88655*x^4 - 58940*x^3 + 2863*x^2 - 4557*x + 30361)
 
gp: K = bnfinit(x^14 + 42*x^12 - 119*x^11 + 553*x^10 - 2702*x^9 + 6174*x^8 - 15735*x^7 + 36995*x^6 - 61572*x^5 + 88655*x^4 - 58940*x^3 + 2863*x^2 - 4557*x + 30361, 1)
 

Normalized defining polynomial

\( x^{14} + 42 x^{12} - 119 x^{11} + 553 x^{10} - 2702 x^{9} + 6174 x^{8} - 15735 x^{7} + 36995 x^{6} - 61572 x^{5} + 88655 x^{4} - 58940 x^{3} + 2863 x^{2} - 4557 x + 30361 \)

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:  \(-104770985911247257875546875=-\,5^{7}\cdot 7^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.21$
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 Galois and abelian over $\Q$.
Conductor:  \(245=5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{245}(1,·)$, $\chi_{245}(34,·)$, $\chi_{245}(36,·)$, $\chi_{245}(69,·)$, $\chi_{245}(71,·)$, $\chi_{245}(104,·)$, $\chi_{245}(106,·)$, $\chi_{245}(139,·)$, $\chi_{245}(141,·)$, $\chi_{245}(174,·)$, $\chi_{245}(176,·)$, $\chi_{245}(209,·)$, $\chi_{245}(211,·)$, $\chi_{245}(244,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{79} a^{12} - \frac{19}{79} a^{11} + \frac{2}{79} a^{10} + \frac{36}{79} a^{9} + \frac{15}{79} a^{8} + \frac{36}{79} a^{7} + \frac{28}{79} a^{6} + \frac{28}{79} a^{5} + \frac{34}{79} a^{4} + \frac{24}{79} a^{3} - \frac{11}{79} a^{2} - \frac{20}{79} a - \frac{9}{79}$, $\frac{1}{38773564319412795975206859460829} a^{13} - \frac{111637784316912571557299686320}{38773564319412795975206859460829} a^{12} - \frac{4643714966374025504520652759732}{38773564319412795975206859460829} a^{11} + \frac{7621234676510962077734429201864}{38773564319412795975206859460829} a^{10} - \frac{2431455967135637793865922512699}{38773564319412795975206859460829} a^{9} - \frac{9172361265689719770421569027349}{38773564319412795975206859460829} a^{8} + \frac{11132302808433319256076261153644}{38773564319412795975206859460829} a^{7} - \frac{17279032022943626042989270832102}{38773564319412795975206859460829} a^{6} + \frac{3386938453365023177976895630882}{38773564319412795975206859460829} a^{5} + \frac{13119972141149482932433582522505}{38773564319412795975206859460829} a^{4} - \frac{4714945864507167865844109040169}{38773564319412795975206859460829} a^{3} - \frac{17401800844317668327168047002426}{38773564319412795975206859460829} a^{2} - \frac{10367868341882617507866620485840}{38773564319412795975206859460829} a - \frac{3700030579107906441080816332982}{38773564319412795975206859460829}$

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

Class group and class number

$C_{254}$, which has order $254$ (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:  \( 35256.6897369 \) (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

\(\Q(\sqrt{-35}) \), 7.7.13841287201.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$5$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
7Data not computed