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Properties

Label 14.14.1282006464...8125.1
Degree $14$
Signature $[14, 0]$
Discriminant $5^{7}\cdot 71^{12}$
Root discriminant $86.35$
Ramified primes $5, 71$
Class number $1$ (GRH)
Class group Trivial (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![42725, 155950, -546085, 220050, 357294, -249558, -53449, 69297, -2735, -7505, 992, 331, -58, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 58*x^12 + 331*x^11 + 992*x^10 - 7505*x^9 - 2735*x^8 + 69297*x^7 - 53449*x^6 - 249558*x^5 + 357294*x^4 + 220050*x^3 - 546085*x^2 + 155950*x + 42725)
 
gp: K = bnfinit(x^14 - 5*x^13 - 58*x^12 + 331*x^11 + 992*x^10 - 7505*x^9 - 2735*x^8 + 69297*x^7 - 53449*x^6 - 249558*x^5 + 357294*x^4 + 220050*x^3 - 546085*x^2 + 155950*x + 42725, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 58 x^{12} + 331 x^{11} + 992 x^{10} - 7505 x^{9} - 2735 x^{8} + 69297 x^{7} - 53449 x^{6} - 249558 x^{5} + 357294 x^{4} + 220050 x^{3} - 546085 x^{2} + 155950 x + 42725 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1282006464112563369862578125=5^{7}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.35$
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:  $5, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(355=5\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{355}(1,·)$, $\chi_{355}(261,·)$, $\chi_{355}(321,·)$, $\chi_{355}(329,·)$, $\chi_{355}(174,·)$, $\chi_{355}(304,·)$, $\chi_{355}(179,·)$, $\chi_{355}(116,·)$, $\chi_{355}(214,·)$, $\chi_{355}(119,·)$, $\chi_{355}(314,·)$, $\chi_{355}(91,·)$, $\chi_{355}(316,·)$, $\chi_{355}(101,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{10} + \frac{2}{25} a^{9} - \frac{2}{25} a^{8} - \frac{2}{5} a^{7} - \frac{9}{25} a^{6} + \frac{4}{25} a^{5} - \frac{2}{5} a^{4} + \frac{4}{25} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{25} a^{11} - \frac{1}{25} a^{9} - \frac{1}{25} a^{8} - \frac{4}{25} a^{7} + \frac{12}{25} a^{6} - \frac{3}{25} a^{5} + \frac{4}{25} a^{4} + \frac{2}{25} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{2125} a^{12} - \frac{42}{2125} a^{11} - \frac{7}{425} a^{10} + \frac{138}{2125} a^{9} + \frac{1}{2125} a^{8} - \frac{159}{425} a^{7} - \frac{521}{2125} a^{6} - \frac{38}{125} a^{5} + \frac{979}{2125} a^{4} - \frac{43}{425} a^{3} + \frac{128}{425} a^{2} + \frac{23}{85} a + \frac{9}{85}$, $\frac{1}{760661364590501159125} a^{13} - \frac{9724050719732413}{44744786152382421125} a^{12} + \frac{5365179953436091868}{760661364590501159125} a^{11} + \frac{7252534958730794483}{760661364590501159125} a^{10} + \frac{3950793316165230147}{44744786152382421125} a^{9} - \frac{35268539238277469}{760661364590501159125} a^{8} - \frac{301123581322477104056}{760661364590501159125} a^{7} + \frac{287167260042172899088}{760661364590501159125} a^{6} - \frac{12150818096935489122}{760661364590501159125} a^{5} - \frac{330440170432600455016}{760661364590501159125} a^{4} + \frac{36499395744273017788}{152132272918100231825} a^{3} + \frac{1773447216211257854}{8948957230476484225} a^{2} + \frac{49086024436658026}{6085290916724009273} a + \frac{14795629905696307404}{30426454583620046365}$

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:  $13$
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:  \( 2275616720.816886 \) (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

\(\Q(\sqrt{5}) \), 7.7.128100283921.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ R ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$71$71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$
71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$