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Properties

Label 14.0.10557874494...6875.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 7^{7}\cdot 71^{12}$
Root discriminant $228.47$
Ramified primes $5, 7, 71$
Class number $809984$ (GRH)
Class group $[2, 2, 2, 2, 4, 4, 3164]$ (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![314845525, -143854350, 85299995, -22736110, 7172534, -1040258, 272781, -41683, 23535, -4645, 722, 31, 12, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 12*x^12 + 31*x^11 + 722*x^10 - 4645*x^9 + 23535*x^8 - 41683*x^7 + 272781*x^6 - 1040258*x^5 + 7172534*x^4 - 22736110*x^3 + 85299995*x^2 - 143854350*x + 314845525)
 
gp: K = bnfinit(x^14 - 5*x^13 + 12*x^12 + 31*x^11 + 722*x^10 - 4645*x^9 + 23535*x^8 - 41683*x^7 + 272781*x^6 - 1040258*x^5 + 7172534*x^4 - 22736110*x^3 + 85299995*x^2 - 143854350*x + 314845525, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 12 x^{12} + 31 x^{11} + 722 x^{10} - 4645 x^{9} + 23535 x^{8} - 41683 x^{7} + 272781 x^{6} - 1040258 x^{5} + 7172534 x^{4} - 22736110 x^{3} + 85299995 x^{2} - 143854350 x + 314845525 \)

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:  \(-1055787449474652775306737176796875=-\,5^{7}\cdot 7^{7}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $228.47$
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:  $5, 7, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2485=5\cdot 7\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{2485}(1,·)$, $\chi_{2485}(1539,·)$, $\chi_{2485}(1156,·)$, $\chi_{2485}(2309,·)$, $\chi_{2485}(456,·)$, $\chi_{2485}(174,·)$, $\chi_{2485}(1681,·)$, $\chi_{2485}(2451,·)$, $\chi_{2485}(1749,·)$, $\chi_{2485}(1014,·)$, $\chi_{2485}(314,·)$, $\chi_{2485}(1891,·)$, $\chi_{2485}(316,·)$, $\chi_{2485}(2344,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{85} a^{9} - \frac{6}{85} a^{8} + \frac{1}{17} a^{7} - \frac{21}{85} a^{6} + \frac{2}{85} a^{5} - \frac{8}{17} a^{4} - \frac{38}{85} a^{3} - \frac{38}{85} a^{2} - \frac{7}{17} a$, $\frac{1}{425} a^{10} + \frac{2}{425} a^{9} + \frac{8}{425} a^{8} + \frac{31}{85} a^{7} - \frac{64}{425} a^{6} - \frac{211}{425} a^{5} - \frac{41}{85} a^{4} - \frac{121}{425} a^{3} + \frac{24}{85} a^{2} + \frac{12}{85} a$, $\frac{1}{425} a^{11} - \frac{1}{425} a^{9} - \frac{1}{425} a^{8} - \frac{144}{425} a^{7} + \frac{107}{425} a^{6} + \frac{122}{425} a^{5} - \frac{21}{425} a^{4} - \frac{43}{425} a^{3} + \frac{36}{85} a^{2} + \frac{11}{85} a$, $\frac{1}{2125} a^{12} - \frac{2}{2125} a^{11} - \frac{2}{2125} a^{9} - \frac{19}{2125} a^{8} - \frac{48}{425} a^{7} + \frac{32}{125} a^{6} - \frac{231}{2125} a^{5} - \frac{176}{2125} a^{4} + \frac{84}{425} a^{3} - \frac{186}{425} a^{2} - \frac{29}{85} a - \frac{2}{5}$, $\frac{1}{761522844817933392809131555299875} a^{13} + \frac{10312417889550183460665517687}{44795461459878434871125385605875} a^{12} + \frac{437638664280578559824344008463}{761522844817933392809131555299875} a^{11} + \frac{11211443212107480765597152248}{761522844817933392809131555299875} a^{10} + \frac{4381004349620233617117395151394}{761522844817933392809131555299875} a^{9} + \frac{50335953178305728978682213319346}{761522844817933392809131555299875} a^{8} - \frac{200877138832336752240925068454746}{761522844817933392809131555299875} a^{7} + \frac{360982772720553605807141717291558}{761522844817933392809131555299875} a^{6} + \frac{215881874609959275721301005056963}{761522844817933392809131555299875} a^{5} - \frac{372517154479809456205958708934536}{761522844817933392809131555299875} a^{4} + \frac{3172978745798926429314163066878}{152304568963586678561826311059975} a^{3} + \frac{72837706525422354242116490184339}{152304568963586678561826311059975} a^{2} - \frac{10246247353303594289048764188023}{30460913792717335712365262211995} a + \frac{804982174268718753212343795478}{1791818458395137394845015424235}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{3164}$, which has order $809984$ (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:  \( 315114.6966253571 \) (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

\(\Q(\sqrt{-35}) \), 7.7.128100283921.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.14.7.2$x^{14} - 686 x^{8} + 117649 x^{2} - 3294172$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$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}$