↑

Properties

Label 14.0.10781852561...4631.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,13^{7}\cdot 43^{13}$
Root discriminant $118.51$
Ramified primes $13, 43$
Class number $74384$ (GRH)
Class group $[74384]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22606297, 25865442, 27352621, 17768243, 8385937, 3751787, 1195655, 382791, 100892, 16023, 5401, 153, 131, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 131*x^12 + 153*x^11 + 5401*x^10 + 16023*x^9 + 100892*x^8 + 382791*x^7 + 1195655*x^6 + 3751787*x^5 + 8385937*x^4 + 17768243*x^3 + 27352621*x^2 + 25865442*x + 22606297)
 
gp: K = bnfinit(x^14 - x^13 + 131*x^12 + 153*x^11 + 5401*x^10 + 16023*x^9 + 100892*x^8 + 382791*x^7 + 1195655*x^6 + 3751787*x^5 + 8385937*x^4 + 17768243*x^3 + 27352621*x^2 + 25865442*x + 22606297, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 131 x^{12} + 153 x^{11} + 5401 x^{10} + 16023 x^{9} + 100892 x^{8} + 382791 x^{7} + 1195655 x^{6} + 3751787 x^{5} + 8385937 x^{4} + 17768243 x^{3} + 27352621 x^{2} + 25865442 x + 22606297 \)

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:  \(-107818525613017436136763734631=-\,13^{7}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.51$
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:  $13, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(559=13\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{559}(1,·)$, $\chi_{559}(194,·)$, $\chi_{559}(389,·)$, $\chi_{559}(391,·)$, $\chi_{559}(168,·)$, $\chi_{559}(170,·)$, $\chi_{559}(365,·)$, $\chi_{559}(558,·)$, $\chi_{559}(274,·)$, $\chi_{559}(51,·)$, $\chi_{559}(183,·)$, $\chi_{559}(376,·)$, $\chi_{559}(508,·)$, $\chi_{559}(285,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{49} a^{11} - \frac{2}{49} a^{10} + \frac{3}{49} a^{9} + \frac{3}{49} a^{8} - \frac{2}{49} a^{7} + \frac{1}{49} a^{6} - \frac{2}{49} a^{5} + \frac{3}{49} a^{4} - \frac{4}{49} a^{3} - \frac{2}{49} a^{2} + \frac{1}{49} a$, $\frac{1}{3871} a^{12} - \frac{19}{3871} a^{11} + \frac{163}{3871} a^{10} - \frac{209}{3871} a^{9} + \frac{108}{3871} a^{8} - \frac{30}{553} a^{7} - \frac{145}{3871} a^{6} + \frac{359}{3871} a^{5} + \frac{1506}{3871} a^{4} - \frac{1019}{3871} a^{3} + \frac{188}{553} a^{2} + \frac{1677}{3871} a - \frac{12}{79}$, $\frac{1}{6432851577632803430102529746318875433} a^{13} + \frac{244920898402918847625877556543791}{6432851577632803430102529746318875433} a^{12} - \frac{7412483944904065824059617182696892}{6432851577632803430102529746318875433} a^{11} + \frac{251916482595277964646348012719587572}{6432851577632803430102529746318875433} a^{10} - \frac{26396194382980742555269239263928651}{6432851577632803430102529746318875433} a^{9} - \frac{237597297059437077945829119558709499}{6432851577632803430102529746318875433} a^{8} - \frac{210871477845978677914573753954096651}{6432851577632803430102529746318875433} a^{7} - \frac{25417279460607840559492151683785707}{6432851577632803430102529746318875433} a^{6} - \frac{23958822568365840836248224387144356}{173860853449535227840608912062672309} a^{5} - \frac{1310543438715484679791760828198388131}{6432851577632803430102529746318875433} a^{4} + \frac{14166462910997131563600356210899619}{81428500982693714305095313244542727} a^{3} + \frac{23632328584571420216156294024886515}{131282685257812314900051627475895417} a^{2} - \frac{149805120027676883337122781728088907}{6432851577632803430102529746318875433} a - \frac{4787587142149283186336005564722}{10528726061256902309732266218293}$

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

Class group and class number

$C_{74384}$, which has order $74384$ (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:  \( 35991.64185055774 \) (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

\(\Q(\sqrt{-559}) \), 7.7.6321363049.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ R ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$13$13.14.7.1$x^{14} - 43940 x^{8} + 482680900 x^{2} - 250994068$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$43.14.13.11$x^{14} + 205667667$$14$$1$$13$$C_{14}$$[\ ]_{14}$