↑

Properties

Label 14.0.52709004898...2511.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 31^{7}$
Root discriminant $156.47$
Ramified primes $7, 31$
Class number $170688$ (GRH)
Class group $[2, 2, 2, 2, 2, 5334]$ (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![67802507, -38466645, 25812486, -9954511, 4382567, -1269674, 393736, -87041, 22610, -3514, 616, -77, 35, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 35*x^12 - 77*x^11 + 616*x^10 - 3514*x^9 + 22610*x^8 - 87041*x^7 + 393736*x^6 - 1269674*x^5 + 4382567*x^4 - 9954511*x^3 + 25812486*x^2 - 38466645*x + 67802507)
 
gp: K = bnfinit(x^14 - 7*x^13 + 35*x^12 - 77*x^11 + 616*x^10 - 3514*x^9 + 22610*x^8 - 87041*x^7 + 393736*x^6 - 1269674*x^5 + 4382567*x^4 - 9954511*x^3 + 25812486*x^2 - 38466645*x + 67802507, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 35 x^{12} - 77 x^{11} + 616 x^{10} - 3514 x^{9} + 22610 x^{8} - 87041 x^{7} + 393736 x^{6} - 1269674 x^{5} + 4382567 x^{4} - 9954511 x^{3} + 25812486 x^{2} - 38466645 x + 67802507 \)

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:  \(-5270900489883727544021626212511=-\,7^{24}\cdot 31^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.47$
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:  $7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1519=7^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1519}(960,·)$, $\chi_{1519}(1,·)$, $\chi_{1519}(869,·)$, $\chi_{1519}(743,·)$, $\chi_{1519}(652,·)$, $\chi_{1519}(526,·)$, $\chi_{1519}(1394,·)$, $\chi_{1519}(435,·)$, $\chi_{1519}(309,·)$, $\chi_{1519}(1303,·)$, $\chi_{1519}(1177,·)$, $\chi_{1519}(218,·)$, $\chi_{1519}(92,·)$, $\chi_{1519}(1086,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9} - \frac{9}{19} a^{8} + \frac{3}{19} a^{7} + \frac{3}{19} a^{6} + \frac{6}{19} a^{5} - \frac{7}{19} a^{4} - \frac{2}{19} a^{3} - \frac{3}{19} a^{2} - \frac{1}{19} a$, $\frac{1}{19} a^{10} - \frac{2}{19} a^{8} - \frac{8}{19} a^{7} - \frac{5}{19} a^{6} + \frac{9}{19} a^{5} - \frac{8}{19} a^{4} - \frac{2}{19} a^{3} - \frac{9}{19} a^{2} - \frac{9}{19} a$, $\frac{1}{19} a^{11} - \frac{7}{19} a^{8} + \frac{1}{19} a^{7} - \frac{4}{19} a^{6} + \frac{4}{19} a^{5} + \frac{3}{19} a^{4} + \frac{6}{19} a^{3} + \frac{4}{19} a^{2} - \frac{2}{19} a$, $\frac{1}{589} a^{12} + \frac{12}{589} a^{11} + \frac{10}{589} a^{10} - \frac{9}{589} a^{9} - \frac{256}{589} a^{8} - \frac{211}{589} a^{7} - \frac{176}{589} a^{6} + \frac{167}{589} a^{5} + \frac{90}{589} a^{4} + \frac{5}{19} a^{3} + \frac{1}{31} a^{2} - \frac{112}{589} a + \frac{10}{31}$, $\frac{1}{1631019238755153731869657927301081} a^{13} - \frac{275417161507632211953481821251}{1631019238755153731869657927301081} a^{12} + \frac{17441373656376964646352139505462}{1631019238755153731869657927301081} a^{11} + \frac{19228647289255019987793742976193}{1631019238755153731869657927301081} a^{10} - \frac{692047693375846446405156646171}{52613523830811410705472836364551} a^{9} + \frac{159591651936346838056375987213686}{1631019238755153731869657927301081} a^{8} - \frac{83096253577334588408529347755886}{1631019238755153731869657927301081} a^{7} - \frac{531425616888508727358853776697823}{1631019238755153731869657927301081} a^{6} - \frac{410927761919185472302959502290054}{1631019238755153731869657927301081} a^{5} + \frac{241657603645035591465807415050922}{1631019238755153731869657927301081} a^{4} - \frac{739531751874462175109273456113391}{1631019238755153731869657927301081} a^{3} + \frac{155786329538505588630464580041994}{1631019238755153731869657927301081} a^{2} + \frac{268216981687743364560228457111643}{1631019238755153731869657927301081} a + \frac{1884467459518167961316479764371}{85843117829218617466824101436899}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{5334}$, which has order $170688$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-31}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ R ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$7$7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$